ims901

B. Bielefeld (editor)
Conformal Dynamics Problem List.
Abstract: A list of unsolved problems was given at the Conformal Dynamics
Conference which was held at SUNY Stony Brook in November 1989.
Problems were contributed by Ben Bielefeld, Adrien Douady, Curt
McMullen, Jack Milnor, Misuhiro Shishikura, Folkert Tangerman,
and Peter Veerman.

ims902

A. M. Blokh and M. Yu. Lyubich
Measurable Dynamics of SUnimodal Maps of the Interval
Abstract: In this paper we sum up our results on onedimensional measurable dynamics
reducing them to the Sunimodal case (compare Appendix 2). Let f be an Sunimodal map
of the interval having no limit cycles. Then f is ergodic with repect to the Lebesque
measure, and has a unique attractor A in the sense of Milnor. This attractor coincides
with the conservative kernel of f. There are no strongly wandering sets of positive
measure. If f has a finite a.c.i. (absolutely continuous invariant) measure u, then
it has positive entropy. This result is closely related to the following: the measure
of Feigenbaumlike attractors is equal to zero. Some extra topological properties
of Cantor attractors are studied.

ims903

J. J. P. Veerman and F. M. Tangerman
On Aubry Mather Sets
Abstract: Let f be a two dimensional area preserving twist map. For each irrational
rotation number in a certain (non trivial) interval, there is an finvariant
minimal set which preserves order with respect to that rotation number. For large
nonlinearity these sets are, typically, Cantor sets and they are referred to as Aubry
Mather sets.
We prove that under csome assumptions these sets are ordered vertically according
to ascending rotation number ("Monotonicity"). Furthermore, if f statisfies certain
conditions, the right hand points of the gaps in an irrational Cantor set lie on a
single orbit ("Single Gap") and diffusion through these Aubry Mather sets can be
understood as a limit of resonance overlaps (Convergence of Turnstiles). These
conditions essentially establish the existence of a hyperbolic structure and limit the
number of homoclinic minimizing orbits.
Some other results along similar lines are given, such as the continuity at irrational
rotation numbers of the Lyapunov exponent on Aubry Mather sets.

ims904

A.E. Eremenko and M. Yu. Lyubich
Dynamical Properties of Some Classes of Entire Functions
Abstract: The paper is concerned with the dynamics of an entire transcendental function whose inverse has only finitely many singularities. It is rpoven that there are no escaping orbits on the Fatou set. Under some extra assumptions the set of escaping orbits has zero Lebesgue measure. If a function depends analytically on parameters then a periodic point as a function of parameters has only algebraic singularities. This yields the Structural Stability Theorem.

ims905

J. Milnor
Dynamics in One Complex Variable: Introductory Lectures.
Abstract: These notes study the dynamics of iterated holomorphic mappings
from a Riemann surface to itself, concentrating on the
classical case of rational maps of the Riemann sphere.
They are based on introductory lectures given at Stony Brook
during the Fall Term of 198990. These lectures are intended
to introduce the reader to some key ideas in the field, and to
form a basis for further study. The reader is assumed to be
familiar with the rudiments of complex variable theory and of
twodimensional differential geometry.

ims906

J. Milnor
Remarks on Iterated Cubic Maps.
Abstract: This note will discuss the dynamics of iterated cubic maps
from the real or complex line to itself, and will describe the
geography of the parameter space for such maps. It is a rough
survey with few precise statements or proofs, and depends
strongly on work by Douady, Hubbard, Branner and Rees.

ims907

J. J. P. Veerman and F. M. Tangerman
Intersection Properties of Invariant Manifolds in Certain Twist Maps
Abstract: We consider the space N of C^{2} twist maps that satisfy the following requirements. The action is the sum of a purely quadratic term and a periodic potential times a constantk (hereafter called the nonlinearity). The potential restricted to the unit circle is bimodal, i.e. has one local minimum and one local maximum. The following statements are proven for maps in N with nonlinearity k large enough.
The intersection of the unstable and stable invariant manifolds to the hyperbolic minimizing periodic points contains minimizing homoclinic points. Consider two finite pieces of these manifolds that connect two adjacent homoclinic minimizing points (hereafter called fundamental domains). We prove that all such fundamental domains have precisely one point in their intersection (the Single Intersection theorem).
In addition, we show that limit points of minimizing points are recurrent, which implies that Aubry Mather sets (with irrational rotation number) are contained in diamonds formed by local stable and unstable manifolds of nearby minimizing periodic orbits (the Diamond Configuration theorem).
Another corollary concerns the intersection of the minimax orbits with certain symmetry lines of the map.

ims908

J. J. P. Veerman and F. M. Tangerman
Scalings in Circle Maps (I)
Abstract: Let f be a flat spot circle map with irrational rotation number. Located at the edges of the flat spot are nonflat critical points (S: x > Ax^{v} ,v≥1). First, we define scalings associated with the closest returns of the orbit of the critical point. Under the assumption that these scalings go to zero, we prove that the derivative of long iterates of the critical value can be expressed in the scalings. The asymptotic behavior of the derivatives and the scalings can then be calculated. We concentrate on the cases for which one can prove the above assumption. In particular, let one of the singularities be linear. These maps arise for example as the lower bound of the nondecreasing truncations of noninvertible bimodal circle maps. It follows that the derivatives grow at a subexponential rate.

ims909

L. Chen
Shadowing Property for Nondegenerate Zero Entropy Piecewise Monotone Maps
Abstract: Let f be a continuous piecewise monotone map of the interval. If any two periodic orbits of f have different itineraries with respect to the partition of the turning points of f, then f is referred to as "nondegenerate". In this paper we prove that a nondegenerate zero entropy continuous piecewise monotone map f has the Shadowing Property if and only if 1) fdows not have neutral periodic points; 2) for each turning point c of f, either the ωlimit set ω(c,f) of c contains no periodic repellors or every periodic repellor in ω(c,f) is a turning point of f in the orbit of c. As an application of this result, the Shadowing Property for the Feigenbaum map is proven.

ims9010

G. Swiatek
OneDimensional Maps and Poincare Metric
Abstract: Invertible compositions of onedimensional maps are studied
which are assumed to include maps with nonpositive Schwarzian
derivative and others whose sum of distortions is bounded. If
the assumptions of the Koebe principle hold, we show that the
joint distortion of the composition is bounded. On the other
hand, if all maps with possibly nonnegative Schwarzian
derivative are almost linearfractional and their
nonlinearities tend to cancel leaving only a small total, then
they can all be replaced with affine maps with the same domains
and images and the resulting composition is a very good
approximation of the original one. These technical tools are
then applied to prove a theorem about critical circle maps.
(AMS subject code 26A18)

ims9011

J. J. P. Veerman and F. M. Tangerman
Saclings in Circle Maps II
Abstract: In this paper we consider one parameter families of circle maps with nonlinear flat spot singularities. Such circle maps were studied in [Circles I] where in particular we studied the geometry of closest returns to the critical interval for irrational rotation numbers of constant type. In this paper we apply those results to obtain exact relations between scalings in the parameter space to dynamical scalings near parameter values where the rotation number is the golden mean. Then results on [Circles I] can be used to compute the scalings in the parameter space. As far as we are aware, this constitutes the first case in which parameter scalings can be rigorously computed in the presence of highly nonlinear (and nonhyperbolic) dynamics.

ims9012

P. M. Bleher & M. Lyubich
The Julia Sets and Complex Singularities in Hierarchical Ising Models
Abstract: We study the analytical continuation in the complex plane of
free energy of the Ising model on diamondlike hierarchical
lattices. It is known that the singularities of free energy of
this model lie on the Julia set of some rational endomorphism
$f$ related to the action of the MigdalKadanoff renormgroup.
We study the asymptotics of free energy when temperature goes
along hyperbolic geodesics to the boundary of an attractive
basin of $f$. We prove that for almost all (with respect to
the harmonic measure) geodesics the complex critical exponent
is common, and compute it.

ims9013

J. J. P. Veerman and F. M. Tangerman
A Remark on Herman's Theorem for Circle Diffeomorphisms
Abstract: We define a class of real numbers that has full measure and is contained in the set of Roth numbers. We prove the C^{1}  part of Herman's theorem: if f is a C^{3} diffeomorphism of the circle to itself with a rotation number ω in this class, then f is C^{1} conjugate to a rotation by ω. As a result of restrictiing the class of admissible rotation numbers, our proof is substantially shorter than Yoccoz' proof.

ims9014

I. L. R. Goldberg II. L. R. Goldberg and J. Milnor
Fixed Points of Polynomial Maps I. Rotation Sets II. Fixed Point Portraits
Abstract: I. We give a combinatorial analysis of rational rotation subsets of the circle. These are invariant subsets that have welldefined rational rotation numbers under the standard selfcovering maps of S^{1}. This analysis has applications to the classification of dynamical systems generated by polynomicals in one complex variable.
II. Douady, Hubbard and Branner have introduced the concept of a "limb" in the Mandelbrot set. A quadratic map f(z)=z^{2}+c belongs to the p/q limb if and only if there exist q external rays of its Julia set which land at a common fixed point of f, and which are permuted by f with combinatorial rotation number p/q in Q/Z, p/q ≠ 0). (Compare Figure 1 and Appendix C, as well as Lemma 2.2.) This note will make a similar analysis of higher degree polynomials by introducing the concept of the "fixed point portrait" of a monic polynomial map.

ims9015

G. P. Paternain & R. J. Spatzier
New Examples of Manifolds with Completely Integrable Geodesic Flows.
Abstract: We construct Riemannian manifolds with completely integrable
geodesic flows, in particular various nonhomogeneous examples.
The methods employed are a modification of Thimm's method,
Riemannian submersions and connected sums.

ims9016

L. Keen and C. Series
Continuity of Convex Hull Boundaries
Abstract: In this paper we consider families of finitely generated
Kleinian groups {G_{μ}} that depend holomorphically on a
parameter μ which varies in an arbitrary connected domain
in C. The groups G_{μ} are quasiconformally conjugate.
We denote the boundary of the convex hull of the
limit set of G\EC by ∂C{G_{μ}). The quotient ∂C(G_{μ})/G_{μ} is
a union of pleated surfaces each carrying a hyperbolic
structure. We fix our attention on one component S_{μ}
and we address the problem of how it varies with μ. We
prove that both the hyperbolic structure and the bending
measure of the pleating lamination of S_{μ} are continuous
functions of μ.

ims911a

Y. Jiang
Dynamics of Certain Smooth OneDimensional Mappings I: The $C^{1+\alpha}$DenjoyKoebe Distortion Lemma
Abstract: We prove a technical lemma, the $C^{1+\alpha }$DenjoyKoebe
distortion lemma, estimating the distortion of a long
composition of a $C^{1+\alpha }$ onedimensional mapping
$f:M\mapsto M$ with finitely many, nonrecurrent, power law
critical points. The proof of this lemma combines the ideas of
the distortion lemmas of Denjoy and Koebe.

ims911b

Y. Jiang
Dynamics of Certain Smooth OneDimensional Mappings II: Geometrically Finite OneDimensional Mappings
Abstract: We study geometrically finite onedimensional mappings. These
are a subspace of $C^{1+\alpha}$ onedimensional mappings with
finitely many, critically finite critical points. We study
some geometric properties of a mapping in this subspace. We
prove that this subspace is closed under quasisymmetrical
conjugacy. We also prove that if two mappings in this subspace
are topologically conjugate, they are then quasisymmetrically
conjugate. We show some examples of geometrically finite
onedimensional mappings.

ims912

L. Keen and C. Series
Pleating Coordinates for the Maskit Embedding of the Teichmüller Space of Punctured Tori
Abstract: The authors introduce a new set of global parameters, which they call pleating coordinates, for the Teichm\FCller space T_{1,1} of the punctured torus. The space T_{1,1} is embedded as a holomorphic family G_{μ} of Kleinian groups, where the complex parameter μ varies in a simply connected domain M in the complex plane. This is done in such a way that the regular set Ω(G_{μ}) has a unique invariant component Ω_{0}(G_{μ}) and the points in T_{1,1} are represented by the Riemann surface Ω(G_{μ})/G_{μ}. This embedding is in fact the Maskit embedding. The new coordinates are geometric in the sense that they are related to the geometry of the hyperbolic manifold H^{3}/G_{μ}. More precisely, they can be read off from the geometry of the punctured torus ∂C_{0}/G_{μ}, where ∂C_{0} is the component of the convex hull boundary facing Ω_{0}(G_{μ}). The surface ∂C0 has a natural hyperbolic metric and is pleated along geodesics that project to a geodesic lamination λ on ∂C_{0}/G_{μ}. There is some specific choice of transverse measure for the pleating lamination &lamba;, which allows the authors to introduce a notion of pleating length for G_{μ}. The laminations and their pleating lengths are the coordinates for M.

ims913

B. Bielefeld, Y. Fisher, AND J. Hubbard
The Classification of Critically Preperiodic Polynomials as Dynamical Systems
Abstract: The object of this paper is to classify all polynomials p with the properties
that all critical points of p are strictly preperiodic under iteration of p. We
will also characterize the Julia sets of such polynomials.

ims914

M. Rees
A Partial Description of the Parameter Space of Rational Maps of Degree Two: Part 2.
Abstract: This continues the investigation of a combinatorial model for
the variation of dynamics in the family of rational maps of
degree two, by concentrating on those varieties in which one
critical point is periodic. We prove some general results about
nonrational critically finite degree two branched coverings,
and finally identify the boundary of the rational maps in the
combinatorial model, thus completing the proofs of results
announced in Part 1.

ims915

M. Kim & S. Sutherland
Polynomial RootFinding Algorithms and Branched Covers.
Abstract: We construct a family of rootfinding algorithms which exploit
the branched covering structure of a polynomial of degree $d$
with a pathlifting algorithm for finding individual roots. In
particular, the family includes an algorithm that computes an
$\epsilon$factorization of the polynomial which has an
arithmetic complexity of
$\Order{d^2(\log d)^2 + d(\log d)^2\log\epsilon}$.
At the present time (1993), this complexity is the best known
in terms of the degree.

ims916

Y. Jiang, T. Morita, & D. Sullivan
Expanding Direction of the Period Doubling Operator.
Abstract: We prove that the period doubling operator has an expanding
direction at the fixed point. We use the induced operator, a
``PerronFrobenius type operator'', to study the linearization
of the period doubling operator at its fixed point. We then use
a sequence of linear operators with finite ranks to study this
induced operator. The proof is constructive. One can calculate
the expanding direction and the rate of expansion of the period
doubling operator at the fixed point.

ims917

M. Shishikura
The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets.
Abstract: It is shown that the boundary of the Mandelbrot set $M$ has
Hausdorff dimension two and that for a generic $c \in \bM$, the
Julia set of $z \mapsto z^2+c$ also has Hausdorff dimension
two. The proof is based on the study of the bifurcation of
parabolic periodic points.

ims918

J. Graczyk & G. Swiatek
Critical Circle Maps Near Bifurcation.
Abstract: We estimate harmonic scalings in the parameter space of a
oneparameter family of critical circle maps. These estimates
lead to the conclusion that the Hausdorff dimension of the
complement of the frequencylocking set is less than $1$ but not
less than $1/3$. Moreover, the rotation number is a H\"{o}lder
continuous function of the parameter.
AMS subject code: 54H20

ims919

E. Cawley
The Teichmuller Space of an Anosov Diffeomorphism of $T^2$.
Abstract: In this paper we consider the space of smooth conjugacy classes
of an Anosov diffeomorphism of the twotorus. The only
2manifold that supports an Anosov diffeomorphism is the
2torus, and Franks and Manning showed that every such
diffeomorphism is topologically conjugate to a linear example,
and furthermore, the eigenvalues at periodic points are a
complete smooth invariant. The question arises: what sets of
eigenvalues occur as the Anosov diffeomorphism ranges over a
topological conjugacy class? This question can be
reformulated: what pairs of cohomology classes (one determined
by the expanding eigenvalues, and one by the contracting
eigenvalues) occur as the diffeomorphism ranges over a
topological conjugacy class? The purpose of this paper is to
answer this question: all pairs of H\"{o}lder reduced
cohomology classes occur.

ims9110

M. Lyubich
On the Lebesgue Measure of the Julia Set of a Quadratic Polynomial.
Abstract: The goal of this note is to prove the following theorem:
Let $p_a(z) = z^2+a$ be a quadratic polynomial which
has no irrational indifferent periodic points, and is
not infinitely renormalizable. Then the Lebesgue measure
of the Julia set $J(p_a)$ is equal to zero.
As part of the proof we discuss a property of the critical
point to be {\it persistently recurrent}, and relate our
results to corresponding ones for real one dimensional maps.
In particular, we show that in the persistently recurrent case
the restriction $p_a\omega(0)$ is topologically minimal and
has zero topological entropy. The DouadyHubbardYoccoz
rigidity theorem follows this result.

ims9111

M. Lyubich
Ergodic Theory for Smooth OneDimensional Dynamical Systems.
Abstract: In this paper we study measurable dynamics for the widest
reasonable class of smooth one dimensional maps. Three
principle decompositions are described in this class :
decomposition of the global measuretheoretical attractor into
primitive ones, ergodic decomposition and Hopf decomposition.
For maps with negative Schwarzian derivative this was done in
the series of papers [BL1BL5], but the approach to the general
smooth case must be different.

ims9112a

Y. Jiang
Dynamics of Certain Smooth OneDimensional Mappings III: Scaling Function Geometry
Abstract: We study scaling function geometry. We show the existence of the
scaling function of a geometrically finite onedimensional
mapping. This scaling function is discontinuous. We prove that
the scaling function and the asymmetries at the critical points
of a geometrically finite onedimensional mapping form a
complete set of $C^{1}$invariants within a topological
conjugacy class.

ims9112b

Y. Jiang
Dynamics of Certain Smooth OneDimensional Mappings IV: Asymptotic Geometry of Cantor Sets.
Abstract: We study hyperbolic mappings depending on a parameter
$\varepsilon $. Each of them has an invariant Cantor set. As
$\varepsilon $ tends to zero, the mapping approaches the
boundary of hyperbolicity. We analyze the asymptotics of the gap
geometry and the scaling function geometry of the invariant
Cantor set as $\varepsilon $ goes to zero. For example, in the
quadratic case, we show that all the gaps close uniformly with
speed $\sqrt {\varepsilon}$. There is a limiting scaling
function of the limiting mapping and this scaling function has
dense jump discontinuities because the limiting mapping is not
expanding. Removing these discontinuities by continuous
extension, we show that we obtain the scaling function of the
limiting mapping with respect to the Ulamvon Neumann type metric.

ims9113

A. M. Blokh
Periods Implying Almost All Periods, Trees with Snowflakes, and Zero Entropy Maps.
Format: AmSTeX (version 1)
Abstract: Let $X$ be a compact tree, $f$ be a continuous map from $X$ to
itself, $End(X)$ be the number of endpoints and $Edg(X)$ be the
number of edges of $X$. We show that if $n>1$ has no prime
divisors less than $End(X)+1$ and $f$ has a cycle of period
$n$, then $f$ has cycles of all periods greater than
$2End(X)(n1)$ and topological entropy $h(f)>0$; so if $p$ is
the least prime number greater than $End(X)$ and $f$ has cycles
of all periods from $1$ to $2End(X)(p1)$, then $f$ has cycles
of all periods (this verifies a conjecture of Misiurewicz for
tree maps). Together with the spectral decomposition theorem
for graph maps it implies that $h(f)>0$ iff there exists $n$
such that $f$ has a cycle of period $mn$ for any $m$. We also
define {\it snowflakes} for tree maps and show that $h(f)=0$
iff every cycle of $f$ is a snowflake or iff the period of
every cycle of $f$ is of form $2^lm$ where $m\le Edg(X)$ is an
odd integer with prime divisors less than $End(X)+1$.

ims9115

M. Lyubich & J. Milnor
The Fibonacci Unimodal Map.
Abstract: This paper will study topological, geometrical and
measuretheoretical properties of the real Fibonacci map. Our
goal was to figure out if this type of recurrence really gives
any pathological examples and to compare it with the infinitely
renormalizable patterns of recurrence studied by Sullivan. It
turns out that the situation can be understood completely and
is of quite regular nature. In particular, any Fibonacci map
(with negative Schwarzian and nondegenerate critical point)
has an absolutely continuous invariant measure (so, we deal
with a ``regular'' type of chaotic dynamics). It turns out also
that geometrical properties of the closure of the critical
orbit are quite different from those of the Feigenbaum map: its
Hausdorff dimension is equal to zero and its geometry is not
rigid but depends on one parameter.

ims9116

M. Jakobsen & G. Swiatek
Quasisymmetric Conjugacies Between Unimodal Maps.
Abstract: It is shown that some topological equivalency classes of
Sunimodal maps are equal to quasisymmetric conjugacy classes.
This includes some infinitely renormalizable polynomials of
unbounded type.

ims9117

M. Lyubich and A. Volberg
A Comparison of Harmonic and Balanced Measures on Cantor Repellors
Abstract: Let J be a Cantor repellor of a conformal map f. Provided f is a polynomiallike or Rsymmetric, we prove that harmonic measure on J is equivalent to the measure of maximal entropy if and only if f is conformally equivalent to a polynomial. We also show that this is not true for general Cantor repellors: there is a nonpolynomial algebraic function generating a Cantor repellor on which above two measures coincide.

ims9118

B. Bielefeld, S. Sutherland, F. Tangerman, and J.J.P. Veerman
Dynamics of Certain NonConformal Degree Two Maps on the Plane
Abstract: In this paper we consider maps on the plane which are similar
to quadratic maps in that they are degree 2 branched covers of
the plane. In fact, consider for $\alpha$ fixed, maps $f_c$
which have the following form (in polar coordinates):
$$f_c(r\,e^{i\theta})\;=\;r^{2\alpha}\,e^{2i\theta}\,+\,c$$
When $\alpha=1$, these maps are quadratic ($z \maps z^2 + c$),
and their dynamics and bifurcation theory are to some degree
understood. When $\alpha$ is different from one, the dynamics
is no longer conformal. In particular, the dynamics is not
completely determined by the orbit of the critical point.
Nevertheless, for many values of the parameter c, the dynamics
has strong similarities to that of the quadratic family. For
other parameter values the dynamics is dominated by 2
dimensional behavior: saddles and the like.
The objects of study are Julia sets, filledin Julia sets and
the connectedness locus. These are defined in analogy to the
conformal case. The main drive in this study is to see to what
extent the results in the conformal case generalize to that of
maps which are topologically like quadratic maps (and when
$\alpha$ is close to one, close to being quadratic).

ims9119a

Y. Jiang
On the Quasisymmetrical Classification of Infinitely Renormalizable Maps: I. Maps with Feigenbaum's Topology.
Abstract: A semigroup (dynamical system) generated by
$C^{1+\alpha}$contracting mappings is considered. We call a
such semigroup regular if the maximum $K$ of the conformal
dilatations of generators, the maximum $l$ of the norms of the
derivatives of generators and the smoothness $\alpha$ of the
generators satisfy a compatibility condition $K< 1/l^{\alpha}$.
We prove the {\em geometric distortion lemma} for a regular
semigroup generated by $C^{1+\alpha}$contracting mappings.

ims9119b

Y. Jiang
On the Quasisymmetrical Classification of Infinitely Renormalizable Maps: II. Remarks on Maps with a Bounded Type Topology.
Abstract: We use the upper and lower potential functions and Bowen's
formula estimating the Hausdorff dimension of the limit set of
a regular semigroup generated by finitely many
$C^{1+\alpha}$contracting mappings. This result is an
application of the geometric distortion lemma in the first
paper at this series.

ims9120

A. Poirier
On the Realization of Fixed Point Portraits (an addendum to Goldberg & Milnor: Fixed Point Portraits)
Abstract: We establish that every formal critical portrait (as defined
by Goldberg and Milnor), can be realized by a postcritically
finite polynomial.

ims9121

C. Gole
Periodic Orbits for Hamiltonian systems in Cotangent Bundles
Abstract: We prove the existence of at least $cl(M)$ periodic orbits for
certain time dependant Hamiltonian systems on the cotangent
bundle of an arbitrary compact manifold $M$. These Hamiltonians
are not necessarily convex but they satisfy a certain boundary
condition given by a Riemannian metric on $M$. We discretize
the variational problem by decomposing the time 1 map into a
product of ``symplectic twist maps''. A second theorem deals
with homotopically non trivial orbits in manifolds of negative
curvature.

ims9122

Peter Jones
On Removable Sets for Sobolev Spaces in the Plane
Abstract: Let $K$ be a compact subset of $\bar{\bold C} ={\bold R}^2$ and
let $K^c$ denote its complement. We say $K\in HR$, $K$ is
holomorphically removable, if whenever $F:\bar{\bold C}
\to\bar{\bold C}$ is a homeomorphism and $F$ is holomorphic
off $K$, then $F$ is a M\"obius transformation. By composing
with a M\"obius transform, we may assume $F(\infty )=\infty$.
The contribution of this paper is to show that a large class of
sets are $HR$. Our motivation for these results is that these
sets occur naturally (e.g. as certain Julia sets) in dynamical
systems, and the property of being $HR$ plays an important role
in the DouadyHubbard description of their structure.

ims9123

L. Keen, B. Maskit, and C. Series
Geometric Finiteness and Uniqueness for Kleinian Groups with Circle Packing Limit Sets.
Abstract: In this paper, we assume that $G$ is a finitely generated
torsion free nonelementary Kleinian group with $\Omega(G)$
nonempty. We show that the maximal number of elements of $G$
that can be pinched is precisely the maximal number of rank 1
parabolic subgroups that any group isomorphic to $G$ may
contain. A group with this largest number of rank 1 maximal
parabolic subgroups is called {\it maximally parabolic}. We
show such groups exist. We state our main theorems
concisely here.
Theorem I. The limit set of a maximally parabolic group is a
circle packing; that is, every component of its regular set is
a round disc.
Theorem II. A maximally parabolic group is geometrically finite.
Theorem III. A maximally parabolic pinched function group is
determined up to conjugacy in $PSL(2,{\bf C})$ by its abstract
isomorphism class and its parabolic elements.

ims922

J. Graczyk, G. Swiatek, F.M. Tangerman,& J.J.P. Veerman
Scalings in Circle Maps III
Abstract: Circle maps with a flat spot are studied which are
differentiable, even on the boundary of the flat spot.
Estimates on the Lebesgue measure and the Hausdorff dimension
of the nonwandering set are obtained. Also, a sharp transition
is found from degenerate geometry similar to what was found
earlier for nondifferentiable maps with a flat spot to bounded
geometry as in critical maps without a flat spot.

ims923

J. Milnor (appendix by A. Poirier)
Hyperbolic Components in Spaces of Polynomial Maps
Abstract: We consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or
more generally polynomial maps from a finite union of copies of
$\C$ to itself which have degree two or more on each copy. In
any space $\p^{S}$ of suitably normalized maps of this type, the
postcritically bounded maps form a compact subset $\cl^{S}$
called the connectedness locus, and the hyperbolic maps in
$\cl^{S}$ form an open set $\hl^{S}$ called the hyperbolic
connectedness locus. The various connected components
$H_\alpha\subset \hl^{S}$ are called hyperbolic components. It
is shown that each hyperbolic component is a topological cell,
containing a unique postcritically finite map which is called
its center point. These hyperbolic components can be separated
into finitely many distinct ``types'', each of which is
characterized by a suitable reduced mapping schema $\bar S(f)$.
This is a rather crude invariant, which depends only on the
topology of $f$ restricted to the complement of the Julia set.
Any two components with the same reduced mapping schema are
canonically biholomorphic to each other. There are similar
statements for real polynomial maps, or for maps with marked
critical points.

ims924

E. Cawley
The Teichm\"uller Space of the Standard Action of $SL(2,Z)$ on $T^2$ is Trivial.
Abstract: The group $SL(n,{\bf Z})$ acts linearly on $\R^n$, preserving
the integer lattice $\Z^{n} \subset \R^{n}$. The induced
(left) action on the ntorus $\T^{n} = \R^{n}/\Z^{n}$ will be
referred to as the ``standard action''.
It has recently been shown that the standard action of
$SL(n,\Z)$ on $\T^n$, for $n \geq 3$, is both topologically
and smoothly rigid. That is, nearby actions in the space of
representations of $SL(n,\Z)$ into ${\rm Diff}^{+}(\T^{n})$ are
smoothly conjugate to the standard action. In fact, this
rigidity persists for the standard action of a subgroup of
finite index. On the other hand, while the $\Z$ action on
$\T^{n}$ defined by a single hyperbolic element of $SL(n,\Z)$
is topologically rigid, an infinite dimensional space of smooth
conjugacy classes occur in a neighborhood of the linear
action.
The standard action of $SL(2, \Z)$ on $\T^2$ forms an
intermediate case, with different rigidity properties from
either extreme. One can construct continuous deformations of
the standard action to obtain an (arbritrarily near) action to
which it is not topologically conjugate. The purpose of the
present paper is to show that if a nearby action, or more
generally, an action with some mild Anosov properties, is
conjugate to the standard action of $SL(2, \Z)$ on $\T^2$ by a
homeomorphism $h$, then $h$ is smooth. In fact, it will be
shown that this rigidity holds for any noncyclic subgroup of
$SL(2, \Z)$.

ims925

Y. Jiang
Dynamics of certain nonconformal semigroups
Abstract: A semigroup generated by two dimensional $C^{1+\alpha}$
contracting maps is considered. We call a such semigroup
regular if the maximum $K$ of the conformal dilatations of
generators, the maximum $l$ of the norms of the derivatives of
generators and the smoothness $\alpha$ of the generators
satisfy a compatibility condition $K< 1/l^{\alpha}$. We prove
that the shape of the image of the core of a ball under any
element of a regular semigroup is good (bounded geometric
distortion like the Koebe $1/4$lemma \cite{a}). And we use it
to show a lower and a upper bounds of the Hausdorff dimension
of the limit set of a regular semigroup. We also consider a
semigroup generated by higher dimensional maps.

ims926

F. Przytycki and F. Tangerman
Cantor Sets in the Line: Scaling Function and the Smoothness of the Shift Map.
Abstract: Consider $d$ disjoint closed subintervals of the unit interval
and consider an orientation preserving expanding map which maps
each of these subintervals to the whole unit interval. The set
of points where all iterates of this expanding map are defined
is a Cantor set. Associated to the construction of this Cantor
set is the scaling function which records the infinitely deep
geometry of this Cantor set. This scaling function is an
invariant of $C^1$ conjugation. We solve the inverse problem
posed by Dennis Sullivan: given a scaling function, determine
the maximal possible smoothness of any expanding map which
produces it.

ims927

B. Bielefeld and M. Lyubich
Problems in Holomorphic Dynamics
Abstract: This preprint will be published by SpringerVerlag as a chapter
of {\sl Linear and Complex Analysis Problem Book}
(eds. V.~P.~Havin and N.~K.~Nikolskii).
1. Quasiconformal Surgery and Deformations
Ben Bielefeld: Questions in Quasiconformal Surgery
Curt McMullen: Rational maps and Teichm\"uller space
John Milnor: Problem: Thurston's algorithm without
critical finiteness
Mary Rees: A Possible Approach to a Complex Renormalization
Problem
2. Geometry of Julia Sets
Lennart Carleson: Geometry of Julia sets.
John Milnor: Problems on local connectivity.
3. Measurable Dynamics
Mikhail Lyubich: Measure and Dimension of Julia Sets.
Feliks Przytycki: On Invariant Measures for Iterations
of Holomorphic Maps
4. Iterates of Entire Functions
Robert Devaney: Open Questions in NonRational Complex
Dynamics
A. Eremenko and M. Lyubich: Wandering Domains for
Holomorphic Maps
5. Newton's Method
Scott Sutherland: Bad Polynomials for Newton's Method

ims928

E. Bedford, M. Lyubich, and J. Smillie
Polynomial Diffeomorphisms of C^2, IV: The Measure of Maximal Entropy and Laminar Currents
Abstract: This paper concerns the dynamics of polynomial automorphisms
of ${\bf C}^2$. One can associate to such an automorphism
two currents $\mu^\pm$ and the equilibrium measure
$\mu=\mu^+\wedge\mu^$. In this paper we study some
geometric and dynamical properties of these objects. First,
we characterize $\mu$ as the unique measure of maximal
entropy. Then we show that the measure $\mu$ has a local
product structure and that the currents $\mu^\pm$ have a
laminar structure. This allows us to deduce information
about periodic points and heteroclinic intersections. For
example, we prove that the support of $\mu$ coincides with
the closure of the set of saddle points. The methods used
combine the pluripotential theory with the theory of
nonuniformly hyperbolic dynamical systems.

ims929

J. Graczyk and G. Swiatek
Singular Measures in Circle Dynamics.
Abstract: Critical circle homeomorphisms have an invariant measure
totally singular with respect to the Lebesgue measure. We prove
that singularities of the invariant measure are of Holder
type. The Hausdorff dimension of the invariant measure is less
than 1 but greater than 0.

ims9211

J. Milnor
Local Connectivity of Julia Sets: Expository Lectures
Abstract: The following notes provide an introduction to recent work of
Branner, Hubbard and Yoccoz on the geometry of polynomial Julia
sets. They are an expanded version of lectures given in Stony
Brook in Spring 1992. I am indebted to help from the
audience.
Section 1 describes unpublished work by J.C. Yoccoz on local
connectivity of quadratic Julia sets. It presents only the
``easy'' part of his work, in the sense that it considers only
nonrenormalizable polynomials, and makes no effort to describe
the much more difficult arguments which are needed to deal with
local connectivity in parameter space. It is based on second
hand sources, namely Hubbard [Hu1] together with lectures by
Branner and Douady. Hence the presentation is surely quite
different from that of Yoccoz.
Section 2 describes the analogous arguments used by Branner and
Hubbard [BH2] to study higher degree polynomials for which all
but one of the critical orbits escape to infinity. In this
case, the associated Julia set \[J\] is never locally
connected. The basic problem is rather to decide when \[J\] is
totally disconnected. This BrannerHubbard work came before
Yoccoz, and its technical details are not as difficult.
However, in these notes their work is presented simply as
another application of the same geometric ideas.
Chapter 3 complements the Yoccoz results by describing a family
of examples, due to Douady and Hubbard (unpublished), showing
that an infinitely renormalizable quadratic polynomial may have
nonlocallyconnected Julia set. An Appendix describes needed
tools from complex analysis, including the Gr\"otzsch
inequality.

ims9212

A. Poirier
Hubbard Forests
Abstract: The theory of Hubbard trees provides an effective classification
of nonlinear postcritically finite polynomial maps from \C to
itself. This note will extend this classification to the case
of maps from a finite union of copies of \C to itself. Maps
which are postcritically finite and nowhere linear will be
characterized by a ``forest'', which is made up out of one tree
in each copy of \C.

ims9213

P. Boyland
Weak Disks of Denjoy Minimal Sets.
Abstract: This paper investigates the existence of Denjoy minimal sets
and, more generally, strictly ergodic sets in the dynamics of
iterated homeomorphisms. It is shown that for the full
twoshift, the collection of such invariant sets with the weak
topology contains topological balls of all finite dimensions.
One implication is an analogous result that holds for
diffeomorphisms with transverse homoclinic points. It is also
shown that the union of Denjoy minimal sets is dense in the
twoshift and that the set of unique probability measures
supported on these sets is weakly dense in the set of all
shiftinvariant, Borel probability measures.

ims9214

J. Milnor
Remarks on Quadratic Rational Maps
Abstract: This will is an expository description of quadratic rational maps.
Sections 2 through 6 are concerned with the geometry and
topology of such maps. Sections 710 survey of some topics
from the dynamics of quadratic rational maps. There are few
proofs. Section 9 attempts to explore and picture moduli space
by means of complex onedimensional slices. Section 10
describes the theory of real quadratic rational maps.
For convenience in exposition, some technical details have been
relegated to appendices: Appendix A outlines some classical
algebra. Appendix B describes the topology of the space of
rational maps of degree \[d\]. Appendix C outlines several
convenient normal forms for quadratic rational maps, and
computes relations between various invariants.\break Appendix D
describes some geometry associated with the curves
\[\Per_n(\mu)\subset\M\]. Appendix E describes totally
disconnected Julia sets containing no critical points.
Finally, Appendix F, written in collaboration with Tan Lei,
describes an example of a connected quadratic Julia set for
which no two components of the complement have a common
boundary point.

ims9215

C. Gole
Optical Hamiltonians and Symplectic Twist Maps
Abstract: This paper concentrates on optical Hamiltonian systems of
$T*\T^n$, i.e. those for which $\Hpp$ is a positive definite
matrix, and their relationship with symplectic twist maps. We
present theorems of decomposition by symplectic twist maps and
existence of periodic orbits for these systems. The novelty of
these results resides in the fact that no explicit asymptotic
condition is imposed on the system. We also present a theorem
of suspension by Hamiltonian systems for the class of
symplectic twist map that emerges in our study. Finally, we
extend our results to manifolds of negative curvature.

ims9217

M. Martens
Distortion Results and Invariant Cantor Sets of Unimodal Maps.
Abstract: A distortion theory is developed for $S$unimodal maps. It will
be used to get some geometric understanding of invariant Cantor
sets. In particular attracting Cantor sets turn out to have
Lebesgue measure zero. Furthermore the ergodic behavior of
$S$unimodal maps is classified according to a distortion
property, called the Markovproperty.

ims9218

M. Lyubich
Combinatorics, Geometry and Attractors of QuasiQuadratic Maps.
Abstract: The Milnor problem on onedimensional attractors is solved for
Sunimodal maps with a nondegenerate critical point c. It
provides us with a complete understanding of the possible
limit behavior for Lebesgue almost every point. This theorem
follows from a geometric study of the critical set $\omega(c)$
of a "nonrenormalizable" map. It is proven that the scaling
factors characterizing the geometry of this set go down to 0 at
least exponentially. This resolves the problem of the
nonlinearity control in small scales. The proofs strongly
involve ideas from renormalization theory and holomorphic
dynamics.

ims931

E. Bedford, M. Lyubich, and J. Smillie
Distribution of Periodic Points of Polynomial Diffeomorphisms of $C^2$
Abstract: (under construction)
This paper deals with the dynamics of a simple family of
holomorphic diffeomorphisms of $\C^2$: the polynomial
automorphisms. This family of maps has been studied by a
number of authors. We refer to [BLS] for a general introduction
to this class of dynamical systems. An interesting object from
the point of view of potential theory is the equilibrium
measure $\mu$ of the set $K$ of points with bounded orbits. In
[BLS] $\mu$ is also characterized dynamically as the unique
measure of maximal entropy. Thus $\mu$ is also an equilibrium
measure from the point of view of the thermodynamical
formalism. In the present paper we give another dynamical
interpretation of $\mu$ as the limit distribution of the
periodic points of $f$.

ims932

A. Connes, D. Sullivan, N. Teleman
Quasiconformal mappings, operators on Hilbert space, and local formulae for characteristic classes
Abstract: Local formulae are given for the characteristic classes of a quasiconformal manifold using the subspace of exact forms in the Hilbert space of middle dimensional forms. The method applies to combinatorial manifolds and all topological manifolds except certain ones in dimension four.

ims933

Feliks Przytycki
Accessability of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps
Abstract: We prove that if A is the basin of immediate attraction to a
periodic attracting or parabolic point for a rational map f on
the Riemann sphere, if $A$ is completely invariant (i.e.
$f^{1}(A)=A$), and if $\mu$ is an arbitrary $f$invariant
measure with positive Lyapunov exponents on the boundary of
$A$, then $\mu$almost every point $q$ in the boundary of $A$
is accessible along a curve from $A$. In fact we prove the
accessability of every "good" $q$ i.e. such $q$ for which
"small neighbourhoods arrive at large scale" under iteration of
$f$. This generalizes DouadyEremenkoLevinPetersen theorem on
the accessability of periodic sources.

ims934

Feliks Przytycki and Anna Zdunik
Density of periodic sources in the boundary of a basin of attraction for iteration of holomorphic maps, geometric coding trees technique
Abstract: We prove that if A is the basin of immediate attraction to a
periodic attracting or parabolic point for a rational map f on
the Riemann sphere, then periodic points in the boundary of A
are dense in this boundary. To prove this in the non simply
connected or parabolic situations we prove a more abstract,
geometric coding trees version.

ims935

Alfredo Poirier
On Postcritically Finite Polynomials, Part 1: Critical Portraits.
Abstract: We extend the work of Bielefeld, Fisher and Hubbard on Critical
Portraits to the case of arbitrary postcritically finite
polynomials. This determines an effective classification of
postcritically finite polynomials as dynamical systems.
This paper is the first in a series of two based on the
author's thesis, which deals with the classification of
postcritically finite polynomials. In this first part we
conclude the study of critical portraits initiated by Fisher
and continued by Bielefeld, Fisher and Hubbard.

ims936

F. Lalonde & D. McDuff
The Geometry of Symplectic Energy
Abstract: One of the most striking early results in symplectic topology
is Gromov's "NonSqueezing Theorem", which says that it is
impossible to embed a large ball symplectically into a thin
cylinder of the form $\R^{2n} \times B^2$, where $B^2$ is a
$2$disc. This led to Hofer's discovery of symplectic
capacities, which give a way of measuring the size of subsets
in symplectic manifolds. Recently, Hofer found a way to
measure the size (or energy) of symplectic diffeomorphisms by
looking at the total variation of their generating
Hamiltonians. This gives rise to a biinvariant (pseudo)norm
on the group $\Ham(M)$ of compactly supported Hamiltonian
symplectomorphisms of the manifold $M$. The deep fact is that
this pseudonorm is a norm; in other words, the only
symplectomorphism on $M$ with zero energy is the identity map.
Up to now, this had been proved only for sufficiently nice
symplectic manifolds, and by rather complicated analytic
arguments.
In this paper we consider a more geometric version of this
energy, which was first considered by Eliashberg and Hofer in
connection with their study of the extent to which the interior
of a region in a symplectic manifold determines its boundary.
We prove, by a simple geometric argument, that both versions of
energy give rise to genuine norms on all symplectic manifolds.
Roughly speaking, we show that if there were a
symplectomorphism of $M$ which had "too little" energy, one
could embed a large ball into a thin cylinder $M \times B^2$.
Thus there is a direct geometric relation between symplectic
rigidity and energy.
The second half of the paper is devoted to a proof of the
NonSqueezing theorem for an arbitrary manifold $M$. We do not
need to restrict to manifolds in which the theory of
pseudoholomorphic curves behaves well. This is of interest
since most other deep results in symplectic topology are
generalised from Euclidean space to other manifolds by using
this theory, and hence are still not known to be valid for
arbitrary symplectic manifolds.

ims937

Alfredo Poirier
On Postcritically Finite Polynomials, Part 2: Hubbard Trees.
Abstract: We provide an effective classification of postcritically finite
polynomials as dynamical systems by means of Hubbard Trees.
This can be viewed as an application of the results developed
in part 1 (ims935).

ims938

J. Graczyk & G. Swiatek
Induced Expansion for Quadratic Polynomials.
Abstract: We prove that nonhyperbolic nonrenormalizable quadratic
polynomials are expansion inducing. For renormalizable
polynomials a counterpart of this statement is that in the
case of unbounded combinatorics renormalized mappings become
almost quadratic. Technically, this follows from the decay of
the box geometry. Specific estimates of the rate of this decay
are shown which are sharp in a class of Sunimodal mappings
combinatorially related to rotations of bounded type. We use
real methods based on crossratios and Schwarzian derivative
complemented by complexanalytic estimates in terms of
conformal moduli.

ims939

Mikhail Lyubich
Geometry of Quadratic Polynomials: Moduli, Rigidity and Local Connectivity.
Abstract: A key problem in holomorphic dynamics is to classify complex
quadratics $z\mapsto z^2+c$ up to topological conjugacy. The
Rigidity Conjecture would assert that any nonhyperbolic
polynomial is topologically rigid, that is, not topologically
conjugate to any other polynomial. This would imply density of
hyperbolic polynomials in the complex quadratic family (Compare
Fatou [F, p. 73]). A stronger conjecture usually abbreviated as
MLC would assert that the Mandelbrot set is locally connected.
A while ago MLC was proven for quasihyperbolic points by
Douady and Hubbard, and for boundaries of hyperbolic components
by Yoccoz. More recently Yoccoz proved MLC for all at most
finitely renormalizable parameter values. One of our goals is
to prove MLC for some infinitely renormalizable parameter
values. Loosely speaking, we need all renormalizations to have
bounded combinatorial rotation number (assumption C1) and
sufficiently high combinatorial type (assumption C2).
For real quadratic polynomials of bounded combinatorial type
the complex a priori bounds were obtained by Sullivan. Our
result complements the Sullivan's result in the unbounded case.
Moreover, it gives a background for Sullivan's renormalization
theory for some bounded type polynomials outside the real line
where the problem of a priori bounds was not handled before for
any single polynomial. An important consequence of a priori
bounds is absence of invariant measurable line fields on the
Julia set (McMullen) which is equivalent to quasiconformal
(qc) rigidity. To prove stronger topological rigidity we
construct a qc conjugacy between any two topologically
conjugate polynomials (Theorem III). We do this by means of a
pullback argument, based on the linear growth of moduli and a
priori bounds. Actually the argument gives the stronger
combinatorial rigidity which implies MLC.
We complete the paper with an application to the real
quadratic family. Here we can give a precise dichotomy (Theorem
IV): on each renormalization level we either observe a big
modulus, or essentially bounded geometry. This allows us to
combine the above considerations with Sullivan's argument for
bounded geometry case, and to obtain a new proof of the
rigidity conjecture on the real line (compare McMullen and
Swiatek).

ims9310

Philip Boyland
Isotopy Stability of Dynamics on Surfaces.
Abstract: This paper investigates dynamics that persist under isotopy in
classes of orientationpreserving homeomorphisms of orientable
surfaces. The persistence of periodic points with respect to
periodic and strong Nielsen equivalence is studied. The
existence of a dynamically minimal representative with respect
to these relations is proved and necessary and sufficient
conditions for the isotopy stability of an equivalence class
are given. It is also shown that most the dynamics of the
minimal representative persist under isotopy in the sense that
any isotopic map has an invariant set that is semiconjugate to
it.

ims9311

Silvina P. Dawson, Roza Galeeva, John Milnor, & Charles Tresser
A Monotonicity Conjecture for Real Cubic Maps.
Abstract: This is an outline of work in progress. We study the conjecture
that the topological entropy of a real cubic map depends
``monotonely'' on its parameters, in the sense that each locus
of constant entropy in parameter space is a connected set.
This material will be presented in more detail in a later paper.

ims9312

Mikhail Lyubich
Teichmuller space of Fibonacci maps
Abstract: According to Sullivan, a space ${\cal E}$ of unimodal maps with
the same combinatorics (modulo smooth conjugacy) should be
treated as an infinitelydimensional Teichm\"{u}ller space.
This is a basic idea in Sullivan's approach to the
Renormalization Conjecture. One of its principle ingredients
is to supply ${\cal E}$ with the Teichm\"{u}ller metric. To
have such a metric one has to know, first of all, that all maps
of ${\cal E}$ are quasisymmetrically conjugate. This was
proved [Ji] and [JS] for some classes of nonrenormalizable
maps (when the critical point is not too recurrent). Here we
consider a space of nonrenormalizable unimodal maps with in
a sense fastest possible recurrence of the critical point
(called Fibonacci). Our goal is to supply this space with the
Teichm\"{u}ller metric.

ims941

J.H. Hubbard & R. ObersteVorth
Henon Mappings in the Complex Domain II: Projective and Inductive Limits of Polynomials.
Abstract: Let H: $C^2 > C^2$ be the Henon mapping given by
(x,y) > (p(x)  ay,x).
The key invariant subsets are K_+/, the sets of points with
bounded forward images, J_+/ = the boundary of K_+/,
J = the union of J_+ and J_, and K = the union of K_+ and
K_. In this paper we identify the topological structure of
these sets when p is hyperbolic and a is sufficiently small,
ie, when H is a small perturbation of the polynomial p. The
description involves projective and inductive limits of objects
defined in terms of p alone.

ims942

H. Bruin, G. Keller, T. Nowicki, & S. van Strien
Absorbing Cantor sets in dynamical systems: Fibonacci maps
Abstract: In this paper we shall show that there exists a polynomial
unimodal map f: [0,1] > [0,1] which is
1) nonrenormalizable(therefore for each x from a
residual set, $\omega(x)$ is equal to an interval),
2) for which $\omega(c)$ is a Cantor set, and
3) for which $\omega(x)=\omega(c)$ for Lebesgue almost all x.
So the topological and the metric attractor of such a map do
not coincide. This gives the answer to a question posed by
Milnor.

ims943

T. Nowicki and S. van Strien
Polynomial Maps with a Julia Set of Positive Measure
Abstract: In this paper we shall show that there exists L_0
such that for each even integer L >= L_0
there exists $c_1 \in \rz$ for which the Julia set of
$z > z^L + c_1$ has positive Lebesgue measure.
This solves an old problem.
Editor's note: In 1997, it was shown by Xavier Buff that there
was a serious flaw in the argument, leaving a gap in the
proof. Currently (1999), the question of polynomials with
a positive measure Julia sets remains open.

ims944

Roza Galeeva,Marco Martens, & Charles Tresser
Inducing, Slopes, and Conjugacy Classes
Abstract: We show that the conjugacy class of an eventually
expanding continuous piecewise affine interval map is contained
in a smooth codimension 1 submanifold of parameter space. In
particular conjugacy classes have empty interior. This is
based on a study of the relation between induced Markov maps
and ergodic theoretical behavior.

ims945

C. Bishop & P. Jones
Hausdorff dimension and Kleinian groups
Abstract: Let G be a nonelementary, finitely generated Kleinian group,
Lambda(G) its limit set and Omega(G) = S \ Lambda(G)
(S = the sphere)
its set of discontinuity. Let delta(G) be the critical
exponent for the Poincare series and let Lambda_c be
the conical limit set of G.
Suppose Omega_0 is a simply connected component of Omega(G).
We prove that
(1) delta(G) = dim(Lambda_c).
(2) A simply connected component Omega
is either a disk or dim(Omega)>1$.
(3) Lambda(G) is either totally disconnected,
a circle or has dimension > 1,
(4) G is geometrically infinite iff dim(Lambda)=2.
(5) If G_n \to G algebraically then
dim(Lambda) <= \liminf dim(Lambda_n).
(6) The Minkowski dimension of Lambda equals the
Hausdorff dimension.
(7) If Area(Lambda)=0 then delta(G) = dim(Lambda(G)).
The proof also shows that \dim(Lambda(G)) > 1 iff the
conical limit set has dimension > 1 iff the Poincare
exponent of the group is > 1. Furthermore, a simply
connected component of Omega(G) either is a disk or has
nondifferentiable boundary in the the sense that the
(inner) tangent points of \partial Omega have zero
1dimensional measure. Almost every point (with
respect to harmonic measure) is a twist point.

ims946

F. Przytycki
Iterations of Rational Functions: Which Hyperbolic Components Contain Polynomials?
Abstract: Let $H^d$ be the set of all rational maps of degree $d\ge 2$
on the Riemann sphere which are expanding on Julia set. We
prove that if $f\in H^d$ and all or all but one critical points
(or values) are in the immediate basin of attraction to an
attracting fixed point then there exists a polynomial in the
component $H(f)$ of $H^d$ containing $f$. If all critical
points are in the immediate basin of attraction to an
attracting fixed point or parabolic fixed point then $f$
restricted to Julia set is conjugate to the shift on the
onesided shift space of $d$ symbols.
We give exotic examples of maps of an arbitrary degree $d$ with
a nonsimply connected, completely invariant basin of
attraction and arbitrary number $k\ge 2$ of critical points in
the basin. For such a map $f\in H^d$ with $k

ims947

J. Kwapisz
A Toral Diffeomorphism with a NonPolygonal Rotation Set.
Abstract: We construct a diffeomorphism of the twodimensional torus
which is isotopic to the identity and whose rotation set is not
a polygon.

ims948

A. Epstein, L. Keen, & C. Tresser
The Set of Maps F_{a,b}: x > x+a+{b/{2 pi}} sin(2 pi x) with any Given Rotation Interval is Contractible.
Abstract: Consider the twoparameter family of real analytic maps
$F_{a,b}:x \mapsto x+ a+{b\over 2\pi} \sin(2\pi x)$ which are
lifts of degree one endomorphisms of the circle. The purpose of
this paper is to provide a proof that for any closed interval
$I$, the set of maps $F_{a,b}$ whose rotation interval is $I$,
form a contractible set.

ims949

T. Bedford & A. Fisher
Ratio Geometry, Rigidity and the Scenery Process for Hyperbolic Cantor Sets
Abstract: Given a $C^{1+\gamma}$ hyperbolic Cantor set $C$, we
study the sequence $C_{n,x}$ of Cantor subsets which nest down
toward a point $x$ in $C$. We show that $C_{n,x}$ is
asymptotically equal to an ergodic Cantor set valued process.
The values of this process, called limit sets, are indexed by a
H\"older continuous setvalued function defined on D.
Sullivan's dual Cantor set. We show the limit sets are
themselves $C^{k+\gamma}, C^\infty$ or $C^\omega$ hyperbolic
Cantor sets, with the highest degree of smoothness which occurs
in the $C^{1+\gamma}$ conjugacy class of $C$. The proof
of this leads to the following rigidity theorem: if two
$C^{k+\gamma}, C^\infty$ or $C^\omega$ hyperbolic Cantor sets
are $C^1$conjugate, then the conjugacy (with a different
extension) is in fact already $C^{k+\gamma}, C^\infty$ or
$C^\omega$.
Within one $C^{1+\gamma}$ conjugacy class, each smoothness
class is a Banach manifold, which is acted on by the semigroup
given by rescaling subintervals. Conjugacy classes nest down,
and contained in the intersection of them all is a compact set
which is the attractor for the semigroup: the collection of
limit sets. Convergence is exponentially fast, in the $C^1$
norm.

ims9410

A. Poirier
Coexistence of Critical Orbit Types in SubHyperbolic Polynomial Maps
Abstract: We establish necessary and sufficient conditions for the
realization of mapping schemata as postcritically finite
polynomials, or more generally, as postcritically finite
polynomial maps from a finite union of copies of the complex
numbers {\bf C} to itself which have degree two or more in each
copy. As a consequence of these results we prove a
transitivity relation between hyperbolic components in
parameter space which was conjectured by Milnor.

ims9411

Y. Minsky
Extremal Length Estimates and Product Regions in Teichmuller Space
Abstract: We study the Teichm\"uller metric on the Teichm\"uller space of
a surface of finite type, in regions where the injectivity
radius of the surface is small. The main result is that in such
regions the Teichm\"uller metric is approximated up to bounded
additive distortion by the sup metric on a product of lower
dimensional spaces. The main technical tool in the proof is the
use of estimates of extremal lengths of curves in a surface
based on the geometry of their hyperbolic geodesic
representatives.

ims9412

T. Kruger, L. D. Pustyl'nikov, & S. E. Troubetzkoy
Acceleration of Bouncing Balls in External Fields
Abstract: We introduce two models, the FermiUlam model in an external
field and a one dimensional system of bouncing balls in an
external field above a periodically oscillating plate. For
both models we investigate the possibility of unbounded motion.
In a special case the two models are equivalent.

ims9413

P. Boyland
Dual Billiards, Twist Maps, and Impact Oscillators.
Abstract: In this paper techniques of twist map theory are applied to the
annulus maps arising from dual billiards on a strictly convex
closed curve G in the plane. It is shown that there do not
exist invariant circles near G when there is a point on G where
the radius of curvature vanishes or is discontinuous. In
addition, when the radius of curvature is not $C^1$ there are
examples with orbits that converge to a point of G. If the
derivative of the radius of curvature is bounded, such orbits
cannot exist. The final section of the paper concerns an impact
oscillator whose dynamics are the same as a dual billiards map.
The appendix is a remark on the connection of the inverse
problems for invariant circles in billiards and dual billiards.

ims9414

M. Boshernitzan, G. Galperin, T. Kruger, & S. Troubetzkoy
Some Remarks on Periodic Billiard Orbits in Rational Polygons
Abstract: A polygon is called rational if the angle between each pair of
sides is a rational multiple of $\pi.$ The main theorem we
will prove is
Theorem 1: For rational polygons, periodic points of the
billiard flow are dense in the phase space of the billiard flow.
This is a strengthening of Masur's theorem, who has shown
that any rational polygon has ``many'' periodic billiard
trajectories; more precisely, the set of directions of the
periodic trajectories are dense in the set of velocity
directions $\S^1.$ We will also prove some refinements of
Theorem 1: the ``well distribution'' of periodic orbits in the
polygon and the residuality of the points $q \in Q$ with a
dense set of periodic directions.

ims9415

C. LeBrun
Einstein Metrics and Mostow Rigidity
Abstract: Using the new diffeomorphism invariants of Seiberg and Witten,
a uniqueness theorem is proved for Einstein metrics on compact
quotients of irreducible 4dimensional symmetric spaces of
noncompact type. The proof also yields a Riemannian version of
the MiyaokaYau inequality.

ims9416

Y. Minsky
QuasiProjections in Teichmuller Space
Abstract: We consider a geometric property of the closestpoints
projection to a geodesic in Teichm\"uller space: the projection
is called contracting if arbitrarily large balls away from the
geodesic project to sets of bounded diameter. (This property
always holds in negatively curved spaces.) It is shown here to
hold if and only if the geodesic is precompact, i.e. its image
in the moduli space is contained in a compact set. Some
applications are given, e.g. to stability properties of certain
quasigeodesics in Teichm\"uller space, and to estimates of
translation distance for pseudoAnosov maps.

ims9417

M. Martens & C. Tresser
Forcing of Periodic Orbits for Interval Maps and Renormalization of Piecewise Affine Maps
Abstract: We prove that for continuous maps on the interval, the
existence of an ncycle, implies the existence of n1 points
which interwind the original ones and are permuted by the map.
We then use this combinatorial result to show that piecewise
affine maps (with no zero slope) cannot be infinitely
renormalizable.

ims9418

N. I. Chernov & S. Troubetzkoy
Measures with Infinite Lyapunov Exponents for the Periodic Lorentz Gas
Abstract: In \cite{Ch91a} it was shown that the billiard ball map for the
periodic Lorentz gas has infinite topological entropy. In this
article we study the set of points with infinite Lyapunov
exponents. Using the cell structure developed in
\cite{BSC90,Ku} we construct an ergodic invariant probability
measure with infinite topological entropy supported on this
set. Since the topological entropy is infinite this is a
measure of maximal entropy. From the construction it is clear
that there many such measures can coexist on a single component
of topological transitivity. We also construct an ergodic
invariant probability measure with finite entropy which is
supported on this set showing that infinite exponents do not
necessarily lead to infinite entropy.

ims9419

E. Lau and D. Schleicher
Internal Addresses in the Mandelbrot Set and Irreducibility of Polynomials.
Abstract: For the polynomials $p_c(z)=z^d+c$, the periodic points of
periods dividing $n$ are the roots of the polynomials
$P_n(z)=p_c^{\circ n}(z)z$, where any degree $d\geq 2$ is
fixed. We prove that all periodic points of any exact period
$k$ are roots of the same irreducible factor of $P_n$ over
$\cz(c)$. Moreover, we calculate the Galois groups of these
irreducible factors and show that they consist of all
permutations of periodic points which commute with the
dynamics. These results carry over to larger families of maps,
including the spaces of general degree$d$polynomials and
families of rational maps.
Main tool, and second main result, is a combinatorial
description of the structure of the Mandelbrot set and its
degree$d$counterparts in terms of internal addresses
of hyperbolic components. Internal addresses interpret kneading
sequences of angles in a geometric way and answer Devaney's
question: ``How can you tell where in the Mandelbrot a
given rational external ray lands, without having Adrien Douady
at your side?''

ims9420

M. Lyubich & Y. Minsky
Laminations in Holomorphic Dynamics
Abstract: We suggest a way to associate to a rational map of the Riemann
sphere a three dimensional object called a hyperbolic orbifold
3lamination. The relation of this object to the map is
analogous to the relation of a hyperbolic 3manifold to a
Kleinian group. In order to construct the 3lamination we
analyze the natural extension of a rational map and the complex
affine structure on the canonical 2dimensional leaf space
contained in it. In this paper the construction is carried out
in full for postcritically finite maps. We show that the
corresponding laminations have a compact convex core.
As a first application we give a threedimensional proof of
Thurston's rigidity for postcritically finite mappings, via
the "lamination extension" of the proofs of the Mostow and
Marden rigidity and isomorphism theorems for hyperbolic
3manifolds. An Ahlforstype argument for zero measure of the
Julia set is applied along the way. This approach also
provides a new point of view on the Lattes deformable examples.

ims951

D. Gale, J. Propp, S. Sutherland, and S. Troubetzkoy
Further Travels with my Ant
Abstract: We discuss some properties of a class of cellular automata
sometimes called a "generalized ant". This system is perhaps
most easily understood by thinking of an ant which moves about
a lattice in the plane. At each vertex (or "cell"), the ant
turns right or left, depending on the the state of the cell,
and then changes the state of the cell according to certain
prescribed rule strings. (This system has been the subject of
several Mathematical Entertainments columns in the Mathematical
Intelligencer; this article will be a future such column). At
various times, the distributions of the states of the cells for
certain ants is bilaterally symmetric; we categorize a class of
ants for which this is the case and give a proof using Truchet
tiles.

ims952

J. Kiwi
Nonaccessible Critical Points of Cremer Polynomials
Abstract: It is shown that a polynomial with a Cremer periodic point
has a nonaccessible critical point in its Julia set provided
that the Cremer periodic point is approximated by small cycles.

ims953a

F. Lalonde and D. McDuff
Hofer's $L^{\infty}$geometry: energy and stability of Hamiltonian flows, part I
Abstract: Consider the group $\Ham^c(M)$ of compactly supported
Hamiltonian symplectomorphisms of the symplectic manifold
$(M,\om)$ with the Hofer $L^{\infty}$norm. A path in
$\Ham^c(M)$ will be called a geodesic if all sufficiently short
pieces of it are local minima for the Hofer length functional
$\Ll$. In this paper, we give a necessary condition for a
path $\ga$ to be a geodesic. We also develop a necessary
condition for a geodesic to be stable, that is, a local minimum
for $\Ll$. This condition is related to the existence of
periodic orbits for the linearization of the path, and so
extends Ustilovsky's work on the second variation formula.
Using it, we construct a symplectomorphism of $S^2$ which
cannot be reached from the identity by a shortest path. In
later papers in this series, we will use holomorphic methods to
prove the sufficiency of the condition given here for the
characterisation of geodesics as well as the sufficiency of the
condition for the stability of geodesics. We will also
investigate conditions under which geodesics are absolutely
lengthminimizing.

ims953b

F. Lalonde and D. McDuff
Hofer's $L^{\infty}$geometry: energy and stability of Hamiltonian flows, part II
Abstract: In this paper we first show that the necessary condition
introduced in our previous paper is also a sufficient condition
for a path to be a geodesic in the group $\Ham^c(M)$ of
compactly supported Hamiltonian symplectomorphisms. This
applies with no restriction on $M$. We then discuss conditions
which guarantee that such a path minimizes the Hofer length.
Our argument relies on a general geometric construction (the
gluing of monodromies) and on an extension of Gromov's
nonsqueezing theorem both to more general manifolds and to
more general capacities. The manifolds we consider are
quasicylinders, that is spaces homeomorphic to $M \times D^2$
which are symplectically ruled over $D^2$. When we work with
the usual capacity (derived from embedded balls), we can prove
the existence of paths which minimize the length among all
homotopic paths, provided that $M$ is semimonotone. (This
restriction occurs because of the wellknown difficulty with
the theory of $J$holomorphic curves in arbitrary $M$.)
However, we can only prove the existence of lengthminimizing
paths (i.e. paths which minimize length amongst {\it all}
paths, not only the homotopic ones) under even more restrictive
conditions on $M$, for example when $M$ is exact and convex or
of dimension $2$. The new difficulty is caused by the
possibility that there are nontrivial and very short loops in
$\Ham^c(M)$. When such lengthminimizing paths do exist, we
can extend the BialyPolterovich calculation of the Hofer norm
on a neighbourhood of the identity ($C^1$flatness).
Although it applies to a more restricted class of
manifolds, the HoferZehnder capacity seems to be better
adapted to the problem at hand, giving sharper estimates in
many situations. Also the capacityarea inequality for split
cylinders extends more easily to quasicylinders in this case.
As applications, we generalise Hofer's estimate of the time for
which an autonomous flow is lengthminimizing to some manifolds
other than $\R^{2n}$, and derive new results such as the
unboundedness of Hofer's metric on some closed manifolds, and a
linear rigidity result.

ims954

Y. Moriah & J. Schultens
Irreducible Heegaard Splittings of Seifert Fibered Spaces are Either Vertical or Horizontal
Abstract: Irreducible 3manifolds are divided into Haken manifolds and nonHaken manifolds. Much is known about the Haken manifolds and this knowledge has been obtained by using the fact that they contain incompressible surfaces. On the other hand, little is known about nonHaken manifolds. As we cannot make use of incompressible surfaces we are forced to consider other methods for studying these manifolds. For example, exploiting the structure of their Heegaard splittings. This approach is enhanced by the result of Casson and Gordon [CG1] that irreducible Heegaard splittings are either strongly irreducible (see Definition 1.2) or the manifold is Haken. Hence, the study of Heegaard splittings as a mean of understanding 3manifolds, whether they are Haken or not, takes on a new significance.

ims955

G. Levin and S. van Strien
Local Connectivity of the Julia Set of Real Polynomials.
Abstract: One of the main questions in the field of complex dynamics
is the question whether the Mandelbrot set is locally
connected, and related to this, for which maps the Julia set is
locally connected. In this paper we shall prove the following
Main Theorem:
Let $f$ be a polynomial of the form $f(z)=z^d +c$ with
$d$ an even integer and $c$ real. Then the Julia set of
$f$ is either totally disconnected or locally connected.
In particular, the Julia set of $z^2+c$ is locally connected
if $c \in [2,1/4]$ and totally disconnected otherwise.

ims957

J. Hu and D. Sullivan
Topological Conjugacy of Circle Diffeomorphisms
Abstract: The classical criterion for a circle diffeomorphism to be
topologically conjugate to an irrational rigid rotation was
given by A. Denjoy. In 1985, one of us (Sullivan) gave a new
criterion. There is an example satisfying Denjoy's bounded
variation condition rather than Sullivan's Zygmund condition
and vice versa. This paper will give the third criterion which
is implied by either of the above criteria.

ims959

C. Bishop, P. Jones, R. Pemantle, and Y. Peres
The Dimension of the Brownian Frontier is Greater than 1
Abstract: Consider a planar Brownian motion run for finite time. The
frontier or ``outer boundary'' of the path is the boundary of
the unbounded component of the complement. Burdzy (1989)
showed that the frontier has infinite length. We improve this
by showing that the Hausdorff dimension of the frontier is
strictly greater than 1. (It has been conjectured that the
Brownian frontier has dimension $4/3$, but this is still
open.) The proof uses Jones's Traveling Salesman Theorem and
a selfsimilar tiling of the plane by fractal tiles known as
Gosper Islands.

ims9510

R. Canary, Y. Minsky, and E. Taylor
Spectral theory, Hausdorff dimension and the topology of hyperbolic 3manifolds
Abstract: Let $M$ be a compact 3manifold whose interior admits a
complete hyperbolic structure. We let $\Lambda(M)$ be the
supremum of $\lambda_0(N)$ where $N$ varies over all hyperbolic
3manifolds homeomorphic to the interior of $N$. Similarly, we
let $D(M)$ be the infimum of the Hausdorff dimensions of limit
sets of Kleinian groups whose quotients are homeomorphic to
the interior of $M$. We observe that $\Lambda(M)=D(M)(2D(M))$
if $M$ is not handlebody or a thickened torus. We characterize
exactly when $\Lambda(M)=1$ and $D(M)=1$ in terms of the
characteristic submanifold of the incompressible core of $M$.

ims9511

Roberto Silvotti
On a conjecture of Varchenko
Abstract: Varchenko conjectured that, under certain genericity
conditions, the number of critical points of a product $\phi$
of powers of linear functions on $\Bbb C^n$ should be given by
the Euler characteristic of the complement of the divisor of
$\phi$ (i.e., a union of hyperplanes). In this note two
independent proofs are given of a direct generalization of
Varchenko's conjecture to the case of a generalized meromorphic
function on an algebraic manifold whose divisor can be any
(generally singular) hypersurface. The first proof uses
characteristic classes and a formula of GaussBonnet type for
affine algebraic varieties. The second proof uses Morse theory.

ims9512

M. Yampolsky
Complex Bounds for Critical Circle Maps
Abstract: We use the methods developed with M. Lyubich for proving
complex bounds for real quadratics to extend E. De Faria's
complex a priori bounds to all critical circle maps with an
irrational rotation number. The contracting property for
renormalizations of critical circle maps follows.
In the Appendix we give an application of the complex bounds
for proving local connectivity of some Julia sets.

ims9513

J. Hu and C. Tresser
Period Doubling, Entropy, and Renormalization
Abstract: We show that in any family of stunted sawtooth maps, the set
of maps whose set of periods is the set of all powers of 2 has
no interior point, i.e., the combinatorial description of the
boundary of chaos coincides with the topological description.
We also show that, under mild assumptions, smooth multimodal
maps whose set of periods is the set of all powers of 2 are
infinitely renormalizable.

ims9514

M. Lyubich
Dynamics of quadratic polynomials II: Rigidity.
Abstract: This is a continuation of the series of notes on the dynamics
of quadratic polynomials. We show the following
Rigidity Theorem: Any combinatorial class contains at most one
quadratic polynomial satisfying the secondary limbs condition
with apriori bounds.
As a corollary, such maps are combinatorially and topologically
rigid, and as a consequence, the Mandelbrot set is locally
connected at the correspoinding parameter values.

ims961a

P. Boyland and C. Gole
Lagrangian Systems on Hyperbolic Manifolds
Abstract: This paper gives two results that show that the dynamics of
a timeperiodic Lagrangian system on a hyperbolic manifold
are at least as complicated as the geodesic flow of a
hyperbolic metric. Given a hyperbolic geodesic in the
Poincar\'e ball, Theorem A asserts that there are minimizers
of the lift of the Lagrangian system that are a bounded
distance away and have a variety of approximate speeds.
Theorem B gives the existence of a collection of compact
invariant sets of the EulerLagrange flow that are
semiconjugate to the geodesic flow of a hyperbolic metric.
These results can be viewed as a generalization of the
AubryMather theory of twist maps and the
HedlundMorseGromov theory of minimal geodesics on closed
surfaces and hyperbolic manifolds.

ims961b

P. Boyland and C. Gole
Dynamical Stability in Lagrangian Systems
Abstract: This paper surveys various results concerning stability
for the dynamics of Lagrangian (or Hamiltonian) systems on
compact manifolds. The main, positive results state,
roughly, that if the configuration manifold carries a
hyperbolic metric, \ie a metric of constant negative
curvature, then the dynamics of the geodesic flow persists
in the EulerLagrange flows of a large class of
timeperiodic Lagrangian systems. This class contains all
timeperiodic mechanical systems on such manifolds. Many of
the results on Lagrangian systems also hold for twist maps
on the cotangent bundle of hyperbolic manifolds.
We also present a new stability result for autonomous
Lagrangian systems on the two torus which shows, among other
things, that there are minimizers of all rotation
directions. However, in contrast to the previously known
\cite{hedlund} case of just a metric, the result allows the
possibility of gaps in the speed spectrum of minimizers.
Our negative result is an example of an autonomous
mechanical Lagrangian system on the twotorus in which this
gap actually occurs. The same system also gives us an
example of a EulerLagrange minimizer which is not a Jacobi
minimizer on its energy level.

ims962

E. Prado
Teichmuller distance for some polynomiallike maps
Abstract: In this work we will show that the Teichm\"{u}ller distance for
all elements of a certain class of generalized polynomiallike
maps (the class of offcritically hyperbolic generalized
polynomiallike maps) is actually a distance, as in the case of
real polynomials with connected Julia set, as studied by
Sullivan. This class contains several important classes of
generalized polynomiallike maps, namely: Yoccoz, Lyubich,
Sullivan and Fibonacci. In our proof we can not use external
arguments (like external classes). Instead we use hyperbolic
sets inside the Julia sets of our maps. Those hyperbolic sets
will allow us to use our main analytic tool, namely Sullivan's
rigidity Theorem for nonlinear analytic hyperbolic systems.
Lyubich has constructed a measure of maximal entropy measure $m$
on the Julia set of any rational function $f$. Zdunik
classified exactly when the Hausdorff dimension of $m$ equals
the Hausdorff dimension of the Julia set. We show that the
strict inequality holds if $f$ is offcrititcally hyperbolic,
except for Chebyshev polynomials. This result is a particular
case of Zdunik's result if we consider $f$ as a polynomial, but
is an extension of Zdunik's result if $f$ is a generalized
polynomiallike map. The proof follows from the nonexistence
of invariant affine structure.

ims963

M. Martens
The Periodic Points of Renormalization
Abstract: It will be shown that the renormalization operator, acting on
the space of smooth unimodal maps with critical exponent
greater than 1, has periodic points of any combinatorial type.

ims964

L. Wenstrom
Parameter Scaling for the Fibonacci Point
Abstract: We prove geometric and scaling results for the real Fibonacci
parameter value in the quadratic family $f_c(z) = z^2+c$. The
principal nest of the Yoccoz parapuzzle pieces has rescaled
asymptotic geometry equal to the filledin Julia set of
$z^21$. The modulus of two such successive parapuzzle pieces
increases at a linear rate. Finally, we prove a ``hairiness"
theorem for the Mandelbrot set at the Fibonacci point when
rescaling at this rate.

ims965

M. Lyubich
Dynamics of quadratic polynomials, III: Parapuzzle and SBR measures.
Abstract: This is a continuation of notes on dynamics of quadratic
polynomials. In this part we transfer the our prior geometric
result to the parameter plane. To any parameter value c
in the Mandelbrot set (which lies outside of the main cardioid
and little Mandelbrot sets attached to it) we associate a
``principal nest of parapuzzle pieces'' and show that the
moduli of the annuli grow at least linearly.
The main motivation for this work was to prove the following:
Theorem B (joint with Martens and Nowicki). Lebesgue
almost every real quadratic polynomial which is nonhyperbolic
and at most finitely renormalizable has a finite absolutely
continuous invariant measure.

ims966

M. Martens and T. Nowicki
Invariant Measures for Typical Quadratic Maps
Abstract: A sufficient geometrical condition for the existence of
absolutely continuous invariant probability measures for
Sunimodal maps will be discussed. The Lebesgue typical
existence of such measures in the quadratic family will be a
consequence.

ims967

S. Zakeri
On Critical Points of Proper Holomorphic Maps on the Unit Disk.
Abstract: We prove that a proper holomorphic map on the unit disk in the
complex plane is uniquely determined up to postcomposition
with a Moebius transformation by its critical points.

ims968

E. Prado
Ergodicity of conformal measures for unimodal polynomials
Abstract: We show that for any unimodal polynomial $f$ with real
coefficients, all conformal measures for $f$ are ergodic.

ims969

Y. Lyubich
A new advance in the Bernstein Problem in mathematical genetics
Abstract: A S.N.Bernstein problem is solved under a natural
irreducibility condition. Earlier this result was obtained
only in some special case.

ims9610

A. Epstein and M. Yampolsky
Geography of the Cubic Connectedness Locus I: Intertwining Surgery
Abstract: We exhibit products of Mandelbrot sets in the twodimensional
complex parameter space of cubic polynomials. These products
were observed by J. Milnor in computer experiments which
inspired Lavaurs' proof of non localconnectivity for the
cubic connectedness locus. Cubic polynomials in such a product
may be renormalized to produce a pair of quadratic maps. The
inverse construction is an {\it intertwining surgery} on two
quadratics. The idea of intertwining first appeared in a
collection of problems edited by Bielefeld. Using
quasiconformal surgery techniques of Branner and Douady,
we show that any two quadratics may be intertwined to obtain a
cubic polynomial. The proof of continuity in our twoparameter
setting requires further considerations involving ray
combinatorics and a pullback argument.

ims9611

H. Masur and Y. Minsky
Geometry of the complex of curves I: Hyperbolicity
Abstract: The Complex of Curves on a Surface is a simplicial complex
whose vertices are homotopy classes of simple closed curves,
and whose simplices are sets of homotopy classes which can be
realized disjointly. It is not hard to see that the complex is
finitedimensional, but locally infinite. It was introduced
by Harvey as an analogy, in the context of Teichmuller space,
for Tits buildings for symmetric spaces, and has been studied
by Harer and Ivanov as a tool for understanding mapping class
groups of surfaces. In this paper we prove that, endowed with
a natural metric, the complex is hyperbolic in the sense of
Gromov.
In a certain sense this hyperbolicity is an explanation of why
the Teichmuller space has some negativecurvature properties
in spite of not being itself hyperbolic: Hyperbolicity in the
Teichmuller space fails most obviously in the regions
corresponding to surfaces where some curve is extremely short.
The complex of curves exactly encodes the intersection
patterns of this family of regions (it is the "nerve" of the
family), and we show that its hyperbolicity means that the
Teichmuller space is "relatively hyperbolic" with respect to
this family. A similar relative hyperbolicity result is
proved for the mapping class group of a surface.
(revised version of January 1998)

ims9612

M. Martens and W. deMelo
Universal Models for Lorenz Maps
Abstract: The existence of smooth families of Lorenz maps exhibiting all
possible dynamical behavior is established and the structure
of the parameter space of these families is described.

ims9613

E. deFaria
Asymptotic Rigidity of Scaling Ratios for Critical Circle Mappings.
Abstract: In this paper we establish $C^2$ apriori bounds for the
scaling ratios of critical circle mappings in a form that
gives also a compactness property for the renormalization
operator.

ims9614

N. Sidorov and A. Vershik
Egrodic Properties of Erd\"os Measure, the Entropy of the Goldenshift, and Related Problems.
Abstract: We define a twosided analog of Erd\"os measure on the space of
twosided expansions with respect to the powers of the golden
ratio, or, equivalently, the Erd\"os measure on the 2torus. We
construct the transformation (goldenshift) preserving both
Erd\"os and Lebesgue measures on $T^2$ which is the
induced automorphism with respect to the ordinary shift (or the
corresponding Fibonacci toral automorphism) and proves to be
Bernoulli with respect to both measures in question. This
provides a direct way to obtain formulas for the entropy
dimension of the Erd\"os measure on the interval, its entropy
in the sense of GarsiaAlexanderZagier and some other results.
Besides, we study central measures on the Fibonacci graph, the
dynamics of expansions and related questions.

ims971

J.J.P. Veerman
Hausdorff Dimension of Boundaries of SelfAffine Tiles in R^n
Abstract: We present a new method to calculate the Hausdorff dimension of
a certain class of fractals: boundaries of selfaffine tiles.
Among the interesting aspects are that even if the affine
contraction underlying the iterated function system is not
conjugated to a similarity we obtain an upper and and
lowerbound for its Hausdorff dimension. In fact, we obtain the
exact value for the dimension if the moduli of the eigenvalues
of the underlying affine contraction are all equal (this
includes Jordan blocks). The tiles we discuss play an important
role in the theory of wavelets.
We calculate the dimension for a number of examples.

ims972

J.J.P. Veerman and L. Jonker
Rigidity Properties Of Locally Scaling Fractals
Abstract: Local scaling of a set means that in a neighborhood of a point
the structure of the set can be mapped into a finer scale
structure of the set. These scaling transformations are compact
sets of locally affine (that is: with uniformly
$\alpha$H\"older continuous derivatives) contractions. In this
setting, without any assumption on the spacing of these
contractions such as the open set condition, we show that the
measure of the set is an upper semicontinuous of the scaling
transformation in the $C^0$topology. With a restriction on the
'nonconformality' (see below) the Hausdorff dimension is
lower semicontinous function in the $C^{1}$topology. We
include some examples to show that neither of these notions is
continuous.

ims973

P. Le Calvez, M. Martens, C. Tresser, and P. Worfolk
Stably Nonsynchronizable Maps of the Plane
Abstract: Pecora and Carroll presented a notion of synchronization where
an (n1)dimensional nonautonomous system is constructed from a
given $n$dimensional dynamical system by imposing the
evolution of one coordinate. They noticed that the resulting
dynamics may be contracting even if the original dynamics are
not. It is easy to construct flows or maps such that no
coordinate has synchronizing properties, but this cannot be
done in an open set of linear maps or flows in $\R ^n$,
$n\geq 2$. In this paper we give examples of real analytic
homeomorphisms of $\R ^ 2$ such that the nonsynchronizability
is stable in the sense that in a full $C^0$ neighborhood of the
given map, no homeomorphism is synchronizable.

ims974

Andre de Carvalho
Pruning fronts and the formation of horseshoes
Abstract: Let f:E > E be a homeomorphism of the plane E. We define open
sets P, called {\em pruning fronts} after the work of
Cvitanovi\'c, for which it is possible to construct an isotopy
H: E x [0,1] > E with open support contained in the union of
f^{n}(P), such that H(*,0)=f(*) and H(*,1)=f_P(*), where
f_P is a homeomorphism under which every point of P is
wandering. Applying this construction with f being Smale's
horseshoe, it is possible to obtain an uncountable
family of homeomorphisms, depending on infinitely many
parameters, going from trivial to chaotic dynamic behaviour.
This family is a 2dimensional analog of a 1dimensional
universal family.

ims975

J.M. Gambaudo and E. E. Pecou
Dynamical Cocycles with Values in the Artin Braid Group.
Abstract: By considering the way an ntuple of points in the 2disk are
linked together under iteration of an orientation preserving
diffeomorphism, we construct a dynamical cocycle with values in
the Artin braid group. We study the asymptotic properties of
this cocycle and derive a series of topological invariants for
the diffeomorphism which enjoy rich properties.

ims977

B. Hinkle
Parabolic Limits of Renormalization
Abstract: In this paper we give a combinatorial description
of the renormlization limits of infinitely renormalizable
unimodal maps with {\it essentially bounded} combinatorics
admitting quadraticlike complex extensions. As an application
we construct a natural analogue of the perioddoubling
fixed point. Dynamical hairiness is also proven for maps in
this class. These results are proven by analyzing {\it parabolic
towers}: sequences of maps related either by renormalization or
by {\it parabolic renormalization}.

ims978

M. Lyubich
Almost Every Real Quadratic Map is Either Regular or Stochastic
Abstract: We prove uniform hyperbolicity of the renormalization operator
for all possible real combinatorial types. We derive from it
that the set of infinitely renormalizable parameter values in
the real quadratic family $P_c: x\mapsto x^2+c$ has zero
measure. This yields the statement in the title (where
``regular'' means to have an attracting cycle and
``stochastic'' means to have an absolutely continuous
invariant measure). An application to the MLC problem is given.

ims979

A. Epstein
Bounded Hyperbolic Components of Quadratic Rational Maps
Abstract: Let ${\cal H}$ be a hyperbolic component of quadratic rational
maps possessing two distinct attracting cycles. We show that
${\cal H}$ has compact closure in moduli space if and only if
neither attractor is a fixed point.

ims9710

J. Milnor
On Rational Maps with Two Critical Points
Abstract: This is a preliminary investigation of the geometry and
dynamics of rational maps with only two critical points.
(originally titled ``On Bicritical Rational Maps''; revised
April 1999)

ims9711

J. Hubbard, P. Papadopol, and V. Veselov
A Compactification of Henon Mappings in $C^2$ as Dynamical Systems
Abstract: In \cite {HO1}, it was shown that there is a topology on
$\C^2\sqcup S^3$ homeomorphic to a 4ball such that the H\'enon
mapping extends continuously. That paper used a delicate
analysis of some asymptotic expansions, for instance, to
understand the structure of forward images of lines near
infinity. The computations were quite difficult, and it is not
clear how to generalize them to other rational maps.
In this paper we will present an alternative approach,
involving blowups rather than asymptotics. We apply it here
only to H\'enon mappings and their compositions, but the method
should work quite generally, and help to understand the
dynamics of rational maps $f:\Proj^2\ratto\Proj^2$ with points
of indeterminacy. The application to compositions of H\'enon
maps proves a result suggested by Milnor, involving embeddings
of solenoids in $S^3$ which are topologically different from
those obtained from H\'enon mappings.

ims9712

M. Martens and W. de Melo
The Multipliers of Periodic Points in Onedimensional Dynamics
Abstract: It will be shown that the smooth conjugacy class of an
$S$unimodal map which does not have a periodic attractor
neither a Cantor attractor is determined by the multipliers of
the periodic orbits. This generalizes a result by M.Shub and
D.Sullivan for smooth expanding maps of the circle.

ims9713

D. Schleicher
Rational Parameter Rays of the Mandelbrot Set
Abstract: We give a new proof that all external rays of the Mandelbrot
set at rational angles land, and of the relation between the
external angle of such a ray and the dynamics at the landing
point. Our proof is different from the original one, given by
Douady and Hubbard and refined by Lavaurs, in several ways: it
replaces analytic arguments by combinatorial ones; it does not
use complex analytic dependence of the polynomials with respect
to parameters and can thus be made to apply for noncomplex
analytic parameter spaces; this proof is also technically
simpler. Finally, we derive several corollaries about
hyperbolic components of the Mandelbrot set.
Along the way, we introduce partitions of dynamical and
parameter planes which are of independent interest, and we
interpret the Mandelbrot set as a symbolic parameter space of
kneading sequences and internal addresses.

ims9714

K. Keller
Correspondence and Translation Principles for the Mandelbrot set
Abstract: New insights into the combinatorial structure of the the
Mandelbrot set are given by `Correspondence' and `Translation'
Principles both conjectured and partially proved by E. Lau
and D. Schleicher. We provide complete proofs of these
principles and discuss results related to them.
Note: The `Translation' and `Correspondence' Principles
given earlier turned out to be false in the general case.
In April 1999, an errata was added to discuss which parts of
the two statements are incorrect and which parts remain true.

ims9715

J. Kiwi
Rational Rays and Critical Portraits of Complex Polynomials
Abstract: The aim of this work is to describe the equivalence relations
in $\Q/\Z$ that arise as the rational lamination of polynomials
with all cycles repelling. We also describe where in parameter
space one can find a polynomial with all cycles repelling and a
given rational lamination. At the same time we derive some
consequences that this study has regarding the topology of
Julia sets.

ims9716

E. de Faria and W. de Melo
Rigidity of critical circle mappings I
Abstract: We prove that two $C^r$ critical circle maps with the same
rotation number of bounded type are $C^{1+\alpha}$ conjugate
for some $\alpha>0$ provided their successive renormalizations
converge together at an exponential rate in the $C^0$
sense. The number $\alpha$ depends only on the rate of
convergence. We also give examples of $C^\infty$ critical
circle maps with the same rotation number that are not
$C^{1+\beta}$ conjugate for any $\beta>0$.

ims9717

E. de Faria and W. de Melo
Rigidity of critical circle mappings II
Abstract: We prove that any two realanalytic critical circle maps with
cubic critical point and the same irrational rotation number of
bounded type are $C^{1+\alpha}$ conjugate for some $\alpha>0$.

ims9718

X. Buff
Geometry of the Feigenbaum map.
Abstract: We show that the FeigenbaumCvitanovic equation can be
interpreted as a linearizing equation, and the domain of
analyticity of the Feigenbaum fixed point of renormalization as
a basin of attraction. There is a natural decomposition of this
basin which enables to recover a result of local connectivity
by Jiang and Hu for the Feigenbaum Julia set.

ims981a

S. Zakeri
Biaccessiblility in Quadratic Julia Sets I: The LocallyConnected Case
Abstract: Let $f:z\mapsto z^2+c$ be a quadratic polynomial whose Julia
set $J$ is locallyconnected. We prove that the Brolin measure
of the set of biaccessible points in $J$ is zero except when
$f(z)=z^22$ is the Chebyshev quadratic polynomial for which
the corresponding measure is one.

ims981b

D. Schleicher and S. Zakeri
On Biaccessible Points in the Julia set of a Cremer Quadratic Polynomial
Abstract: We prove that the only possible biaccessible points in the
Julia set of a Cremer quadratic polynomial are the Cremer fixed
point and its preimages. This gives a partial answer to a
question posed by C. McMullen on whether such a Julia set can
contain any biaccessible point at all

ims981c

S. Zakeri
Biaccessiblility in Quadratic Julia Sets II: The Siegel and Cremer Cases.
Abstract: Let $f$ be a quadratic polynomial which has an irrationally
indifferent fixed point $\alpha$. Let $z$ be a biaccessible
point in the Julia set of $f$. Then:
1. In the Siegel case, the orbit of $z$ must eventually hit
the critical point of $f$.
2. In the Cremer case, the orbit of $z$ must eventually hit
the fixed point $\alpha$.
Siegel polynomials with biaccessible critical point certainly
exist, but in the Cremer case it is possible that biaccessible
points can never exist.
As a corollary, we conclude that the set of biaccessible points
in the Julia set of a Siegel or Cremer quadratic polynomial has
Brolin measure zero.

ims982

V. Kaimanovich
The Poisson Formula for Groups with Hyperbolic Properties.
Abstract: The Poisson boundary of a group $G$ with a probability measure
$\mu$ on it is the space of ergodic components of the time
shift in the path space of the associated random walk. Via a
generalization of the classical Poisson formula it gives an
integral representation of bounded $\mu$harmonic functions on
$G$. In this paper we develop a new method of identifying the
Poisson boundary based on entropy estimates for conditional
random walks. It leads to simple purely geometric criteria of
boundary maximality which bear hyperbolic nature and allow us
to identify the Poisson boundary with natural topological
boundaries for several classes of groups: word hyperbolic
groups and discontinuous groups of isometries of Gromov
hyperbolic spaces, groups with infinitely many ends, cocompact
lattices in CartanHadamard manifolds, discrete subgroups of
semisimple Lie groups, polycyclic groups, some wreath and
semidirect products including BaumslagSolitar groups.

ims983

E. Bedford and M. Jonsson
Regular Polynomial Endomorphisms of C^k
Abstract: We study the dynamics of polynomial mappings
$f:{\bf C}^k\to{\bf C}^k$ of degree $d\ge2$
that extend continuously to projective space ${\bf P}^k$.
Our approach is to study the dynamics near the hyperplane at
infinity and then making a descent to $K$  the set of points
with bounded orbits  via external rays.

ims984

S. Zakeri
On Dynamics of Cubic Siegel Polynomials
Abstract: Motivated by the work of Douady, Ghys, Herman and Shishikura on
Siegel quadratic polynomials, we study the onedimensional
slice of the cubic polynomials which have a fixed Siegel disk
of rotation number $\theta$, with $\theta$ being a given
irrational number of Brjuno type. Our main goal is to prove
that when $\theta$ is of bounded type, the boundary of the
Siegel disk is a quasicircle which contains one or both
critical points of the cubic polynomial. We also prove that the
locus of all cubics with both critical points on the boundary
of their Siegel disk is a Jordan curve, which is in some sense
parametrized by the angle between the two critical points. A
main tool in the bounded type case is a related space of
degree 5 Blaschke products which serve as models for our
cubics. Along the way, we prove several results about the
connectedness locus of these cubic polynomials.

ims985

M. Yampolsky
The Attractor of Renormalization and Rigidity of Towers of Critical Circle Maps
Abstract: We demonstrate the existence of a global attractor A with a
Cantor set structure for the renormalization of critical circle
mappings. The set A is invariant under a generalized
renormalization transformation, whose action on A is conjugate
to the twosided shift.

ims986

C. Bishop
Nonremovable sets for quasiconformal and locally biLipschitz mappings in R^3
Abstract: We give an example of a totally disconnected set $E \subset
{\Bbb R}^3$ which is not removable for quasiconformal
homeomorphisms, i.e., there is a homeomorphism $f$ of ${\Bbb
R}^3$ to itself which is quasiconformal off $E$, but not
quasiconformal on all of ${\Bbb R}^3$. The set $E$ may be
taken with Hausdorff dimension $2$. The construction also
gives a nonremovable set for locally biLipschitz
homeomorphisms.

ims987

J.J.P. Veerman, M.M. Peixoto, A.C. Rocha, and S. Sutherland
On Brillouin Zones
Abstract: Brillouin zones were introduced by Brillouin in the thirties to
describe quantum mechanical properties of crystals, that is, in
a lattice in $\R^n$. They play an important role in solidstate
physics. It was shown by Bieberbach that Brillouin zones tile
the underlying space and that each zone has the same area. We
generalize the notion of Brillouin Zones to apply to an
arbitrary discrete set in a proper metric space, and show that
analogs of Bieberbach's results hold in this context.
We then use these ideas to discuss focusing of geodesics in
orbifolds of constant curvature. In the particular case of the
Riemann surfaces $\H^2/\Gamma (k)$ (k=2,3, or 5), we
explicitly count the number of geodesics of length $t$ that
connect the point $i$ to itself.

ims988

M. Yampolsky and S. Zakeri
Mating Siegel Quadratic Polynomials
Abstract: Let $F$ be a quadratic rational map of the sphere which has two
fixed Siegel disks with bounded type rotation numbers $\theta$
and $\nu$. Using a new degree 3 Blaschke product model for the
dynamics of $F$ and an adaptation of complex a priori bounds
for renormalization of critical circle maps, we prove that $F$
can be realized as the mating of two Siegel quadratic
polynomials with the corresponding rotation numbers $\theta$
and $\nu$.

ims989

J. Milnor and C. Tresser
On Entropy and Monotonicity for Real Cubic Maps
Abstract: It has been known for some time that the topological entropy is
a nondecreasing function of the parameter in the real quadratic
family, which corresponds to the intuitive idea that more
nonlinearity induces more complex dynamical behavior.
Polynomial families of higher degree depend on several
parameters, so that the very question of monotonicity needs to
be reformulated. For instance, one can say the entropy is
monotone in a multiparameter family if the isentropes, or sets
of maps with the same topological entropy, are connected. Here
we reduce the problem of the connectivity of the isentropes in
the real cubic families to a weak form of the Fatou conjecture
on generic hyperbolicity, which was proved to hold true by
C. Heckman. We also develop some tools which may prove to be
useful in the study of other parameterized families, in
particular a general monotonicity result for stunted sawtooth
maps: the stunted sawtooth family of a given shape can be
understood as a simple family which realizes all the possible
combinatorial structures one can expect with a map of this
shape on the basis of kneading theory. Roughly speaking, our
main result about real cubic families is that they are as
monotone as the stunted sawtooth families with the same shapes
because of Heckman's result (there are two posible shapes for
cubic maps, depending on the behavior at infinity).

ims9810

S. Zakeri
Dynamics of Singular Holomorphic Foliations on the Complex Projective Plane
Abstract: This manuscript is an introduction to the theory of holomorphic
foliations on the complex projective plane. Historically the
subject has emerged from the theory of ODEs in the complex
domain and various attempts to solve Hilbert's 16th Problem,
but with the introduction of complex algebraic geometry,
foliation theory and dynamical systems, it has now become an
interesting subject of its own.

ims9811

J. Kahn
Holomorphic Removability of Julia Sets
Abstract: Let $f(z) = z^2 + c$ be a quadratic polynomial, with c
in the Mandelbrot set. Assume further that both fixed points
of f are repelling, and that f is not renormalizable.
Then we prove that the Julia set J of f is
holomorphically removable in the sense that every homeomorphism
of the complex plane to itself that is conformal off of J
is in fact conformal on the entire complex plane. As a
corollary, we deduce that the Mandelbrot Set is locally
connected at such c.

ims9812

D. Schleicher
On Fibers and Local Connectivity of Compact Sets in C.
Abstract: A frequent problem in holomorphic dynamics is to prove local
connectivity of Julia sets and of many points of the Mandelbrot
set; local connectivity has many interesting implications. The
intention of this paper is to present a new point of view for
this problem: we introduce fibers of these sets, and the goal
becomes to show that fibers are ``trivial'', i.e. they consist
of single points. The idea is to show ``shrinking of puzzle
pieces'' without using specific puzzles. This implies local
connectivity at these points, but triviality of fibers is a
somewhat stronger property than local connectivity. Local
connectivity proofs in holomorphic dynamics often actually
yield that fibers are trivial, and this extra knowledge is
sometimes useful.
Since we believe that fibers may be useful in further
situations, we discuss their properties for arbitrary compact
connected and full sets in the complex plane. This allows to
use them for connected filledin Julia sets of polynomials, and
we deduce for example that infinitely renormalizable
polynomials of the form $z^d+c$ have the property that the
impression of any dynamic ray at a rational angle is a single
point. An appendix reviews known topological properties of
compact, connected and full sets in the plane.
The definition of fibers grew out of a new brief proof that the
Mandelbrot set is locally connected at every Misiurewicz point
and at every point on the boundary of a hyperbolic
component. This proof works also for ``Multibrot sets'', which
are the higher degree cousins of the Mandelbrot set. These sets
are discussed in a selfcontained sequel (IMS Preprint
1998/13a). Finally, we relate triviality of fibers to tuning
and renormalization in IMS Preptint 1998/13b.

ims9813a

D. Schleicher
On Fibers and Local Connectivity of Mandelbrot and Multibrot Sets
Abstract: We give new proofs that the Mandelbrot set is locally connected
at every Misiurewicz point and at every point on the boundary
of a hyperbolic component. The idea is to show ``shrinking of
puzzle pieces'' without using specific puzzles. Instead, we
introduce fibers of the Mandelbrot set and show that fibers of
certain points are ``trivial'', i.e., they consist of single
points. This implies local connectivity at these points.
Locally, triviality of fibers is strictly stronger than local
connectivity. Local connectivity proofs in holomorphic dynamics
often actually yield that fibers are trivial, and this extra
knowledge is sometimes useful. We include the proof that local
connectivity of the Mandelbrot set implies density of
hyperbolicity in the space of quadratic polynomials.
We write our proofs more generally for Multibrot sets,
which are the loci of connected Julia sets for polynomials of
the form $z\mapsto z^d+c$.
Although this paper is a continuation of preprint 1998/12, it
has been written so as to be independent of the discussion of
fibers of general compact connected and full sets in $\C$ given
there.

ims9813b

D. Schleicher
On Fibers and Renormalization of Julia Sets and Multibrot Sets
Abstract: We continue the description of Mandelbrot and Multibrot sets
and of Julia sets in terms of fibers which was begun in
IMS preprints 1998/12 and 1998/13a. The question of local
connectivity of these sets is discussed in terms of fibers and
becomes the question of triviality of fibers. In this paper,
the focus is on the behavior of fibers under renormalization
and other surgery procedures. We show that triviality of fibers
of Mandelbrot and Multibrot sets is preserved under tuning maps
and other (partial) homeomorphisms. Similarly, we show for
unicritical polynomials that triviality of fibers of Julia sets
is preserved under renormalization and other surgery
procedures, such as the BrannerDouady homeomorphisms. We
conclude with various applications about quadratic polynomials
and its parameter space: we identify embedded paths within the
Mandelbrot set, and we show that Petersen's theorem about
quadratic Julia sets with Siegel disks of bounded type
generalizes from period one to arbitrary periods so that they
all have trivial fibers and are thus locally connected.

ims991

A. Epstein
Infinitesimal Thurston Rigidity and the FatouShishikura Inequality
Abstract: We prove a refinement of the FatouShishikura Inequality  that
the total count of nonrepelling cycles of a rational map is
less than or equal to the number of independent infinite
forward critical orbits  from a suitable application of
Thurston's Rigidity Theorem  the injectivity of $If_*$ on
spaces of meromorphic quadratic differentials.

ims992

V. Kaloshin
Generic Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits
Abstract: Consider a compact manifold M of dimension at least 2
and the space of $C^r$smooth diffeomorphisms Diff$^r(M)$. The
classical ArtinMazur theorem says that for a dense subset D
of Diff$^r(M)$ the number of isolated periodic points grows at
most exponentially fast (call it the AM property).
We extend this result and prove that diffeomorphisms having
only hyperbolic periodic points with the AM property are dense
in Diff$^r(M)$. Our proof of this result is much simpler than
the original proof of ArtinMazur.
The second main result is that the AM property is not (Baire)
generic. Moreover, in a Newhouse domain ${\cal N} \subset
\textup{Diff}^r(M)$, an arbitrary quick growth of the number of
periodic points holds on a residual set. This result follows
from a theorem of GonchenkoShilnikovTuraev, a detailed proof
of which is also presented.

ims993

J. Milnor
Periodic Orbits, Externals Rays and the Mandelbrot Set: An Expository Account
Abstract: A key point in Douady and Hubbard's study of the Mandelbrot set
$M$ is the theorem that every parabolic point $c\ne 1/4$ in $M$
is the landing point for exactly two external rays with angles
which are periodic under doubling. This note will try to
provide a proof of this result and some of its consequences
which relies as much as possible on elementary combinatorics,
rather than on more difficult analysis. It was inspired by
section 2 of the recent thesis of Schleicher
(see also IMS preprint 1994/19, with E. Lau), which
contains very substantial simplifications of the DouadyHubbard
proofs with a much more compact argument, and is highly
recommended. The proofs given here are rather different from
those of Schleicher, and are based on a combinatorial study of
the angles of external rays for the Julia set which land on
periodic orbits. The results in this paper are mostly well
known; there is a particularly strong overlap with the work of
Douady and Hubbard. The only claim to originality is in
emphasis, and the organization of the proofs.

ims994

X. Buff and C. Henriksen
Scaling Ratios and Triangles in Siegel Disks
Abstract: Let $f(z)=e^{2i\pi\theta} z+z^2$, where $\theta$ is a quadratic
irrational. McMullen proved that the Siegel disk for $f$ is
selfsimilar about the critical point. We give a lower bound
for the ratio of selfsimilarity, and we show that if
$\theta=(\sqrt 51)/2$ is the golden mean, then there exists a
triangle contained in the Siegel disk, and with one vertex at
the critical point. This answers a 15 year old conjecture.

ims995

K. Pilgrim
Dessins d'enfants and Hubbard Trees
Abstract: We show that the absolute Galois group acts faithfully on the
set of Hubbard trees. Hubbard trees are finite planar trees,
equipped with selfmaps, which classify postcritically finite
polynomials as holomorphic dynamical systems on the complex
plane. We establish an explicit relationship between certain
Hubbard trees and the trees known as ``dessins d'enfant''
introduced by Grothendieck.

ims996

W. de Melo and A. A. Pinto
Rigidity of C^2 Infinitely Renormalizable Unimodal Maps
Abstract: Given $C^2$ infinitely renormalizable unimodal maps $f$ and $g$
with a quadratic critical point and the same bounded
combinatorial type, we prove that they are $C^{1+\alpha}$
conjugate along the closure of the corresponding forward
orbits of the critical points, for some $\alpha>0$.

ims997a

B. Weiss
Preface to "On Actions of Epimorphic Subgroups on Homogeneous Spaces" and "Unique Ergodicity on Compact Homogeneous Spaces"
Abstract: This short note serves as a joint introduction to the papers
``On Actions of Epimorphic Subgroups on Homogeneous Spaces"
by Nimesh Shah and Barak Weiss (Stony Brook IMS preprint 1999/7b)
and ``Unique Ergodicity on Compact Homogeneous Spaces" by Barak
Weiss. For the benefit of the readers who are not experts in
the theory of subgroup actions on homogeneous spaces I have
prefaced the papers with some general remarks explaining and
motivating our results, and the connection between them.
The remarks are organized as a comparison between facts which had
been previously known about the action of the geodesic and
horocycle flow on finitevolume Riemann surfaces  the
simplest nontrivial example that falls into our framework 
and our results on subgroup actions on homogeneous spaces.

ims997b

N. Shah and B. Weiss
On Actions of Epimorphic Subgroups on Homogeneous Spaces
Abstract: We show that
for an inclusion $F
The key ingredient in establishing this result is the study of
the limiting distributions of certain translates of a
homogeneous measure. We show that if in addition $G$ is
generated by unipotent elements then there exists $a\in F$ such
that the following holds: Let $U\subset F$ be the subgroup
generated by all unipotent elements of $F$, $x\in L/\Lambda$,
and $\lambda$ and $\mu$ denote the Haar probability measures on
the homogeneous spaces $\cl{Ux}$ and $\cl{Gx}$, respectively
(cf.~Ratner's theorem). Then $a^n\lambda\to\mu$ weakly as
$n\to\infty$.
We also give an algebraic characterization of algebraic
subgroups $F<\SL_n(\R)$ for which all orbit closures are finite
volume almost homogeneous spaces, namely {\it iff\/} the
smallest observable subgroup of $\SL_n(\R)$ containing $F$ has
no nontrivial characters defined over $\R$.

ims997c

B. Weiss
"Unique Ergodicity on Compact Homogeneous Spaces"
Abstract: Extending results of a number of authors, we prove that if $U$
is the unipotent radical of a solvable epimorphic subgroup of
an algebraic group $G$, then the action of $U$ on $G/\Gamma$ is
uniquely ergodic for every cocompact lattice $\Gamma$ in
$G$. This gives examples of uniquely ergodic and minimal
twodimensional flows on homogeneous spaces of arbitrarily high
dimension. Our main tools are Ratner classification of ergodic
invariant measures for the action of a unipotent subgroup on a
homogeneous space, and a simple lemma (the `Cone Lemma') about
representations of epimorphic subgroups.
(revised version of July 1999)

ims998

E. Lindenstrauss and B. Weiss
On Sets Invariant under the Action of the Diagonal Group
Abstract: We investigate closures of orbits for the action of the group
of diagonal matrices acting on $SL(n,R)/SL(n,Z)$, where
$n \geq 3$. It has been conjectured by Margulis that possible
orbitclosures for this action are very restricted. Lending
support to this conjecture, we show that any orbitclosure
containing a compact orbit is homogeneous. Moreover if $n$ is
prime then any orbit whose closure contains a compact orbit is
either compact itself or dense. This implies a numbertheoretic
result generalizing an isolation theorem of Cassels and
SwinnertonDyer for products of linear forms. We also obtain
similar results for other lattices instead of $SL(n,Z)$,
under a suitable irreducibility hypothesis.

ims999

D. Schleicher and J. Zimmer
Dynamic Rays for Exponential Maps
Abstract: We discuss the dynamics of exponential maps
$z\mapsto \lambda e^z$ from the point of view of dynamic
rays, which have been an important tool for the study of
polynomial maps. We prove existence of dynamic rays with
bounded combinatorics and show that they contain all points
which ``escape to infinity'' in a certain way. We then discuss
landing properties of dynamic rays and show that in many
important cases, repelling and parabolic periodic points are
landing points of periodic dynamic rays. For the case of
postsingularly finite exponential maps, this needs the use of
spider theory.

ims9910

J.J.P. Veerman and B. Stosic
On the Dimensions of Certain Incommensurably Constructed Sets
Abstract: It is well known that the Hausdorff dimension of the invariant
set $\Lambda_t$ of an iterated function system ${\cal F}_t$ on
$\R^n$ depending smoothly on a parameter $t$ does not vary
continuously. In fact, it has been shown recently that in
general it varies lowersemicontinuously. For a specific
family of systems we investigate numerically the conjecture
that discontinuities in the dimension only arise when in some
iterate of the iterated function system two (or more) of its
branches coincide. This happens in a set of codimension one,
but which is dense. All the other points are conjectured to be
points of continuity.

ims0001

J.H. Hubbard and P. Papadopol
Newton's Method Applied to Two Quadratic Equations in $C^2$ Viewed as a Global Dynamical System.
Abstract: In this paper, we will study Newton's method for solving two
simultaneous quadratic equations in two variables. Presumably,
there is no need to motivate a study of Newton's method, in
one or several variables. The algorithm is of immense
importance, and understanding its behavior is of obvious
interest. It is perhaps harder to motivate the case of two
simultaneous quadratic equations in two variables, but this is
the simplest nondegenerate case.

ims0002

A. de Carvalho and T. Hall
Pruning, Kneading and Thurston's Classification of Surface Homeomorphisms.
Abstract: In this paper, new techniques for studying the dynamics of
families of surface homeomorphisms are introduced. Two
dynamical deformation theories are presented  one for
surface homeomorphisms, called pruning, and another for graph
endomorphisms, called kneading  both giving conditions under
which all of the dynamics in an open set can be destroyed,
while leaving the dynamics unchanged elsewhere. These theories
are then used to give a proof of Thurston's classification
theorem for surface homeomorphisms up to isotopy.

ims0003

T. Lundh
In Search of an Evolutionary Coding Style.
Abstract: In the near future, all the human genes will be identified. But
understanding the functions coded in the genes is a much harder
problem. For example, by using block entropy, one has that the
DNA code is closer to a random code then written text, which in
turn is less ordered then an ordinary computer code; see
\cite{schmitt}. Instead of saying that the DNA is badly
written, using our programming standards, we might say that it
is written in a different style  an evolutionary style. We
will suggest a way to search for such a style in a quantified
manner by using an artificial life program, and by giving a
definition of general codes and a definition of style for such
codes.

ims0004

D. Schleicher
Attracting Dynamics of Exponential Maps
Abstract: We give a complete classification of hyperbolic components in
the space of iterated maps $z\mapsto \lambda\exp(z)$, and we
describe a preferred parametrization of those components. This
leads to a complete classification of all exponential maps with
attracting dynamics.

ims0005

F. Loray and J. Rebelo
Stably chaotic rational vector fields on $\Bbb C\Bbb P^n$.
Abstract: We construct an open set $\Cal U$ of rational foliations of
arbitrarily fixed degree $d \ge 2$ by curves in
$\Bbb C\Bbb P^n$ such that any foliation $\Cal F\in\Cal U$
has a finite number of singularities and satisfies the
following chaotic properties.
Minimality: any leaf (curve) is dense in $\Bbb C\Bbb P^n$.
Ergodicity: any Lebesgue measurable subset of leaves has
zero or total Lebesgue measure.
Entropy: the topological entropy is strictly positive
even far from singularities.
Rigidity: if $\Cal F$ is conjugate to some $\Cal F'\in\Cal U$
by a homeomorphism close to the identity, then they are
also conjugate by a projective transformation.
The main analytic tool employed in the construction of these
foliations is the existence of several pseudoflows in the
closure of pseudogroups generated by perturbations of elements
in $\text{Diff}(\Bbb C^n,0)$ on a fixed ball.

ims0006

C. L. Petersen and S. Zakeri
On the Julia Set of a Typical Quadratic Polynomial with a Siegel disk.
Abstract: Let $0< \theta <1$ be an irrational number with continued
fraction expansion $\theta=[a_1, a_2, a_3, \ldots]$, and
consider the quadratic polynomial $\pt : z \mapsto
e^{2\pi i \theta} z +z^2$. By performing a transquasiconformal
surgery on an associated Blaschke product model, we prove that
if $$\log a_n = {\cal O} (\sqrt{n})\ \operatorname{as}\ n
\to \infty ,$$ then the Julia set of $\pt$ is locallyconnected
and has Lebesgue measure zero. In particular, it follows that
for almost every $0< \theta < 1$, the quadratic $\pt$ has
a Siegel disk whose boundary is a Jordan curve
passing through the critical point of $\pt$. By standard
renormalization theory, these results generalize to the
quadratics which have Siegel disks of higher periods.

ims0007

Y. Minsky
Bounded geometry for Kleinian groups.
Abstract: We show that a Kleinian surface group, or hyperbolic 3manifold
with a cusppreserving homotopyequivalence to a surface, has
bounded geometry if and only if there is an upper bound on an
associated collection of coefficients that depend only on its
end invariants. Bounded geometry is a positive lower bound on
the lengths of closed geodesics. When the surface is a
oncepunctured torus, the coefficients coincide with the
continued fraction coefficients associated to the ending
laminations.

ims0008

Y. Minsky and B. Weiss
Nondivergence of Horocyclic Flows on Moduli Space.
Abstract: The earthquake flow and the Teichm\"uller horocycle flow
are flows on bundles over the Riemann moduli space of a
surface, and are similar in many respects to unipotent
flows on homogeneous spaces of Lie groups. In analogy
with results of Margulis, Dani and others in the
homogeneous space setting, we prove strong nondivergence
results for these flows. This extends previous work of
Veech. As corollaries we obtain that every closed invariant
set for the earthquake (resp. Teichm\"uller horocycle)
flow contains a minimal set, and that almost every quadratic
differential on a Teichm\"uller horocycle orbit has a
uniquely ergodic vertical foliation.

ims0009

J. RiveraLetelier
Rational maps with decay of geometry: rigidity, Thurston's algorithm and local connectivity.
Abstract: We study dynamics of rational maps that satisfy a decay of
geometry condition. Well known conditions of nonuniform
hyperbolicity, like summability condition with exponent one,
imply this condition. We prove that Julia sets have zero
Lebesgue measure, when not equal to the whole sphere, and in
the polynomial case every connected component of the Julia set
is locally connected. We show how rigidity properties of
quasiconformal maps that are conformal in a big dynamically
defined part of the sphere, apply to dynamics. For example we
give a partial answer to a problem posed by Milnor about
Thurston's algorithm and we give a proof that the Mandelbrot
set, and its higher degree analogues, are locally connected at
parameters that satisfy the decay of geometry condition.
Moreover we prove a theorem about similarities between the
Mandelbrot set and Julia sets. In an appendix we prove a
rigidity property that extends a key situation encountered by
Yoccoz in his proof of local connectivity of the Mandelbrot set
at at most finitely renormalizable parameters.

ims0010

A. Avila, M. Martens and W. de Melo
On the Dynamics of the Renormalization Operator.
Abstract: An important part of the bifurcation diagram of unimodal maps
corresponds to infinite renormalizable maps. The dynamics of
the renormalization operator describes this part of the
bifurcation pattern precisely. Here we analyze the dynamics of
the renormalization operator acting on the space of $C^k$
infinitely renormalizable maps of bounded type. We prove that
two maps of the same type are exponentially asymptotic. We
suppose $k \geq 3$ and quadratic critical point.

ims0011

F. Ferreira and A. A. Pinto
Explosion of Smoothness from a Point to Everywhere for Conjugacies Between Diffeomorphisms on Surfaces.
Abstract: For diffeomorphisms on surfaces with basic sets, we show the
following type of rigidity result: if a topological conjugacy
between them is differentiable at a point in the basic set then
the conjugacy has a smooth extension to the surface. These
results generalize the similar ones of D. Sullivan, E. de Faria,
and ours for onedimensional expanding dynamics.

ims0101

E. de Faria, W de Melo and A. Pinto
Global Hyperbolicity of Renormalization for $C^r$ Unimodal Mappings.
Abstract: In this paper we extend M.~Lyubich's recent results on the global
hyperbolicity of renormalization of quadraticlike germs to the
space $\mathbb{U}^r$ of $C^r$ unimodal maps with quadratic
critical point. We show that in $\mathbb{U}^r$ the boundedtype
limit sets of the renormalization operator have an invariant
hyperbolic structure provided $r \ge 2+\alpha$ with $\alpha$
close to one. As an intermediate step between Lyubich's results
and ours, we prove that the renormalization operator is
hyperbolic in a Banach space of real analytic maps. We construct
the local stable manifolds and prove that they form a continuous
lamination whose leaves are $C^1$ codimension one Banach
submanifolds of $\mathbb{U}^r$, and whose holonomy is
$C^{1+\beta}$ for some $\beta>0$. We also prove that the global
stable sets are $C^1$ immersed (codimension one) submanifolds as
well, provided $r \ge 3+\alpha$ with $\alpha$ close to one. As a
corollary, we deduce that in generic one parameter families of
$C^r$ unimodal maps, the set of parameters corresponding to
infinitely renormalizable maps of bounded combinatorial type is a
Cantor set with Hausdorff dimension less than one.

ims0102

A. de Carvalho and T. Hall
The Forcing Relation for Horseshoe Braid Types.
Abstract: This paper presents evidence for a conjecture concerning the
structure of the set of braid types of periodic orbits of
Smale's horseshoe map, partially ordered by Boyland's forcing
order. The braid types are partitioned into totally ordered
subsets, which are defined by parsing the symbolic code of a
periodic orbit into two segments, the {\em prefix} and the
{\em decoration}: the set of braid types of orbits with each
given decoration is totally ordered, the order being given by
the unimodal order on symbol sequences. The conjecture is
supported by computer experiment, by proofs of special cases,
and by intuitive argument in terms of pruning theory.

ims0103

G. Birkhoff, M. Martens and C. Tresser
On the scaling structure for period doubling.
Abstract: We describe the order on the ratios that define the generic
universal smooth period doubling Cantor set. We prove that this
set of ratios forms itself a Cantor set, a Conjecture formulated
by Coullet and Tresser in 1977. We also show that the two period
doubling renormalization operators, acting on the codimension one
space of period doubling maps, form an iterated function system
whose limit set contains a Cantor set.

ims0104

A. Epstein and M. Yampolsky
A universal parabolic map.
Abstract: Parabolic renormalization of critical circle maps arises as a
degenerate case of the usual renormalization when the periods
of the renormalized maps become infinite. In the paper we give
new proofs of the main renormalization conjectures for the
parabolic case, which are notably simplier than those required
in the usual case. The title of the paper refers to the
attracting fixed point of the parabolic renormalization, whose
existence we prove.

ims0105

V. Kaimanovich and M. Lyubich
Conformal and Harmonic Measures on Laminations Associated with Rational Maps.
Abstract: In this work we continue the exploration of affine and hyperbolic
laminations associated with rational maps, which were introduced
in \cite{LM}. Our main goal is to construct natural geometric
measures on these laminations: transverse conformal measures on
the affine laminations and harmonic measures on the hyperbolic
laminations. The exponent $\de$ of the transverse conformal
measure does not exceed 2, and is related to the eigenvalue of
the harmonic measure by the formula $\la=\de(\de2)$. In the
course of the construction we introduce a number of geometric
objects on the laminations: the basic cohomology class of an
affine lamination (an obstruction to flatness), leafwise and
transverse conformal streams, the backward and forward Poincar\'e
series and the associated critical exponents. We discuss their
relations to the Busemann and the AnosovSinai cocycles, the
curvature form, currents and transverse invariant measures,
$\la$harmonic functions, PattersonSullivan and Margulis
measures, etc. We also prove that the dynamical laminations in
question are never flat except for several explicit special
cases (rational functions with parabolic Thurston orbifold).

ims0106

H. Miyachi
Cusps in complex boundaries of onedimensional Teichm\"uller space.
Abstract: This paper gives a proof of the conjectural phenomena on the
complex boundary onedimensional slices: Every rational
boundary point is cusp shaped. This paper treats this problem
for Bers slices, the Earle slices, and the Maskit slice.
In proving this, we also obtain the following result: Every
Teichm\"uller modular transformation acting on a Bers slice
can be extended as a quasiconformal mapping on its ambient
space. We will observe some similarity phenomena on the
boundary of Bers slices, and discuss on the dictionary
between Kleinian groups and Rational maps concerning with
these phenomena. We will also give a result related to the
theory of L.Keen and C.Series of pleated varieties in
quasifuchsian space of once punctured tori.

ims0107

F. P. Gardiner, J. Hu and N. Lakic
Earthquake Curves.
Abstract: The first two parts of this paper concern homeomorphisms of
the circle, their associated earthquakes, earthquake laminations
and shearing measures. We prove a finite version of Thurston's
earthquake theorem \cite{Thurston4} and show that it implies
the existence of an earthquake realizing any homeomorphism.
Our approach gives an effective way to compute the lamination.
We then show how to recover the earthquake from the measure,
and give examples to show that locally finite measures on given
laminations do not necessarily yield homeomorphisms. One of
them also presents an example of a lamination ${\cal L}$ and
a measure $\sigma $ such that the corresponding mapping
$h_{\sigma}$ is not a homeomorphism of the circle but
$h_{2\sigma}$ is. The third part of the paper concerns the
dependence between the norm $\sigma _{Th}$ of a measure
$\sigma$ and the norm $h_{cr}$ of its corresponding
quasisymmetric circle homeomorphism $h_{\sigma}$. We first
show that $\sigma _{Th}$ is bounded by a constant multiple
of $h_{cr}$. Conversely, we show for any $C_0>0$, there
exists a constant $C>0$ depending on $C_0$ such that for any
$\sigma $, if $\sigma _{Th}\le C_0$ then
$h_{cr}\le C\sigma _{Th}$. The fourth part of the paper
concerns the differentiability of the earthquake curve
$h_{t\sigma }, t\ge 0,$ on the parameter $t$. We show that for
any locally finite measure $\sigma $, $h_{t\sigma }$ satisfies
the nonautonomous ordinary differential equation
$$\frac{d}{dt} h_{t\sigma}(x)=V_t(h_{t\sigma}(x)), \ t\ge 0,$$
at any point $x$ on the boundary of a stratum of the lamination
corresponding to the measure $\sigma.$ We also show that if the
norm of $\sigma $ is finite, then the differential equation
extends to every point $x$ on the boundary circle, and the
solution to the differential equation an initial condition is
unique. The fifth and last part of the paper concerns
correspondence of regularity conditions on the measure $\sigma,$
on its corresponding mapping $h_{\sigma},$ and on the tangent
vector $$V= V_0 = \frac{d}{dt}\big_{t=0} h_{t\sigma}.$$ We give
equivalent conditions on $\sigma, h_{\sigma}$ and $V$ that
correspond to $h_{\sigma }$ being in {\em Diff}$^ {\ 1+\alpha}$
classes, where $0\le \alpha <1$.

ims0108

J. Hu
Earthquake Measure and Crossratio Distortion.
Abstract: Given an orientationpreserving circle homeomorphism $h$, let
$(E, \mathcal{L})$ denote a Thurston's left or right earthquake
representation of $h$ and $\sigma $ the transversal shearing
measure induced by $(E, \mathcal{L})$. We first show that the
Thurston norm $\cdot _{Th}$ of $\sigma $ is equivalent to
the crossratio distortion norm $\cdot _{cr}$ of $h$, i.e.,
there exists a constant $C>0$ such that
$$\frac{1}{C}h_{cr}\le \sigma _{Th} \le Ch_{cr}$$
for any $h$. Secondly we introduce two new norms on the
crossratio distortion of $h$ and show they are equivalent to
the Thurston norms of the measures of the left and right
earthquakes of $h$. Together it concludes that the Thurston
norms of the measures of the left and right earthquakes of $h$
and the three norms on the crossratio distortion of $h$ are
all equivalent. Furthermore, we give necessary and sufficient
conditions for the measures of the left and right earthquakes
to vanish in different orders near the boundary of the
hyperbolic plane. Vanishing conditions on either measure imply
that the homeomorphism $h$ belongs to certain classes of circle
diffeomorphisms classified by Sullivan in \cite{Sullivan}.

ims0109

J. C. Rebelo and R. R. Silva
The multiple ergodicity of nondiscrete subgroups of ${\rm Diff}^{\omega} ({\mathbb S}^1)$
Abstract: In this work we deal with nondiscrete subgroups of $\dif$,
the group of orientationpreserving analytic diffeomorphisms
of the circle. If $\Gamma$ is such a group, we consider its
natural diagonal action $\ogama$ on the $n$dimensional torus
$\tor^n$. It is then obtained a complete characterization of
these groups $\Gamma$ whose corresponding $\ogama$action on
$\tor^n$ is not piecewise ergodic (cf. Introduction) for all
$n \in \N$ (cf. Theorem~A). Theorem~A can also be interpreted
as an extension of Lie's classification of Lie algebras on
$\s$ to general nondiscrete subgroups of $\s$.

ims0110

A. de Carvalho and T. Hall
How to prune a horseshoe
Abstract: Let $F\colon\ofr^2\to\ofr^2$ be a homeomorphism. An open
$F$invariant subset $U$ of $\ofr^2$ is a {\em pruning region}
for $F$ if it is possible to deform $F$ continuously to a
homeomorphism $F_U$ for which every point of $U$ is wandering,
but which has the same dynamics as $F$ outside of $U$. This
concept was motivated by the {\em Pruning Front Conjecture} of
Cvitanovi\'c, Gunaratne, and Procaccia, which claims that every
H\'enon map can be understood as a pruned horseshoe. This paper
is a survey of pruning theory, concentrating on prunings of the
horseshoe. We describe conditions on a disk $D$ which ensure
that the orbit of its interior is a pruning region; explain how
prunings of the horseshoe can be understood in terms of
underlying tree maps; discuss the connection between pruning and
Thurston's classification theorem for surface homeomorphisms;
motivate a conjecture describing the {\em forcing relation} on
horseshoe braid types; and use this theory to give a precise
statement of the pruning front conjecture.

ims0111

G. Tomanov and B. Weiss
Closed orbits for actions of maximal tori on homogeneous spaces.
Abstract: Let $G$ be a real algebraic group defined over $Q$, let
$\Gamma$ be an arithmetic subgroup, and let $T$ be a maximal
$\R$split torus. We classify the closed orbits for the action
of $T$ on $G/\Gamma,$ and show that they all admit a simple
algebraic description. In particular we show that if $G$ is
reductive, an orbit $Tx\Gamma$ is closed if and only if
$x^{1}Tx$ is defined over $\Q$, and is (totally) divergent if
and only if $x^{1}Tx$ is defined over $\Q$ and $\Q$split. Our
analysis also yields the following: there
is a compact $K \subset G/\Gamma$ which intersects every
$T$orbit. \item if $\Q {\rm rank}(G)<\R{\rm rank}(G)$,
there are no divergent orbits for $T$.

ims0112

J. RiveraLetelier
Espace hyperbolique padique et dynamique des fonctions rationnelles.
Abstract: We study dynamics of rational maps of degree at least 2 with
coefficients in the field $\C_p$, where $p > 1$ is a fixed
prime number. The main ingredient is to consider the action
of rational maps in $p$adic hyperbolic space, denoted $\H_p$.
Hyperbolic space $\H_p$ is provided with a natural distance,
for which it is connected and one dimensional (an $\R$tree).
This advantages with respect to $\C_p$ give new insight into
dynamics; in this paper we prove the following results about
periodic points. In forthcoming papers we give applications to
the Fatou/Julia theory over $\C_p$.
First we prove that the existence of two nonrepelling periodic
points implies the existence of infinitely many of them. This
is in contrast with the complex setting where there can be at
most finitely many nonrepelling periodic points. On the other
hand we prove that every rational map has a repelling fixed
point, either in the projective line or in hyperbolic space.
We also caracterise those rational maps with finitely many
periodic points in hyperbolic space. Such a rational map can
have at most one periodic point (which is then fixed) and we
characterise those rational maps having no periodic points and
those rational maps having precisely one periodic point in
hyperbolic space.
We also prove a formula relating different objects in the
projective line and in hyperbolic space, which are fixed by a
given rational map. Finally we relate hyperbolic space in the
form given here, to well known objects: the BruhatTits building
of $PSL(2, \C_p)$ and the Berkovich space of $\P(\C_p)$.

ims0113

Marco Lenci
Billiards with an infinite cusp
Abstract: Let $f: [0, +\infty) \longrightarrow (0, +\infty)$ be a
sufficiently smooth convex function, vanishing at infinity.
Consider the planar domain $Q$ delimited by the positive
$x$semiaxis, the positive $y$semiaxis, and the graph of $f$.
Under certain conditions on $f$, we prove that the billiard
flow in $Q$ has a hyperbolic structure and, for some examples,
that it is also ergodic. This is done using the cross section
corresponding to collisions with the dispersing part of the
boundary. The relevant invariant measure for this Poincar\'e
section is infinite, whence the need to surpass the existing
results, designed for finitemeasure dynamical systems.

ims0114

Jeremy Tyson
On the conformal dimensions of quasiconvex postcritically finite self similar sets
Abstract: The conformal dimension of a metric space is the infimum of the
Hausdorff dimensions of all quasisymmetrically equivalent
metrics on the space. We show that certain classical selfsimilar
fractal subsets of Euclidean space are not minimal for conformal
dimension by constructing explicit metrics in the quasisymmetry
class of the Euclidean metric with reduced Hausdorff dimension.

ims0115

Artur Avila, Mikhail Lyubich and Welington de Melo
Regular or stochastic dynamics in real analytic families of unimodal maps
Abstract: In this paper we prove that in any nontrivial real analytic
family of unimodal maps, almost any map is either regular (i.e.,
it has an attracting cycle) or stochastic (i.e., it has an
absolutely continuous invariant measure). To this end we show
that the space of analytic maps is foliated by codimensionone
analytic submanifolds, ``hybrid classes''. This allows us to
transfer the regular or stochastic property of the quadratic
family to any nontrivial real analytic family.

ims0201

A. de Carvalho and T. Hall
Braid forcing and starshaped train tracks
Abstract: Global results are proved about the way in which Boyland's forcing
partial order organizes a set of braid types: those of periodic
orbits of Smale's horseshoe map for which the associated train
track is a star. This is a special case of a conjecture
introduced in [1], which claims that forcing organizes all
horseshoe braid types into linearly ordered families which are, in
turn, parametrized by homoclinic orbits to the fixed point of
code 0.

ims0202

S. Zakeri
External rays and the real slice of the mandelbrot set
Abstract: This paper investigates the set of angles of the parameter rays
which land on the real slice [2, 1/4] of the Mandelbrot set. We
prove that this set has zero length but Hausdorff dimension 1.
We obtain the corresponding results for the tuned images of the
real slice. Applications of these estimates in the study of
critically nonrecurrent real quadratics as well as biaccessible
points of quadratic Julia sets are given.

ims0203

J. C. Rebelo
Complete polynomial vector fields on $\C^2$,{\sc Part I}
Abstract: In this work, under a mild assumption, we give the
classification of the complete polynomial
vector fields in two variables up to algebraic automorphisms
of $\C^2$. The general problem is also reduced to the study of
the combinatorics of certain resolutions of singularities.
Whereas we deal with $\C$complete vector fields,
our results also apply to $\R$complete ones thanks to a
theorem of Forstneric [Fo].

ims0204

A. de Carvalho and M. Paternain
Monotone quotients of surface diffeomorphisms
Abstract: A homeomorphism of a compact metric space is {\em tight} provided
every nondegenerate compact connected (not necessarily invariant)
subset carries positive entropy. It is shown that every
$C^{1+\alpha}$ diffeomorphism of a closed surface factors to a
tight homeomorphism of a generalized cactoid (roughly, a surface
with nodes) by a semiconjugacy whose fibers carry zero entropy.

ims0205

S. Zakeri
David maps and Hausdorff Dimension
Abstract: David maps are generalizations of classical planar
quasiconformal maps for which the
dilatation is allowed to tend to infinity in a
controlled fashion. In this note we examine how these
maps distort Hausdorff dimension. We show \vs
\begin{enumerate}
\item[$\bullet$]
Given $\alpha$ and $\beta$ in $[0,2]$, there exists a David
map $\varphi:\CC \to \CC$ and a compact set $\Lambda$ such that
$\Hdim \Lambda =\alpha$ and $\Hdim \varphi(\Lambda)=\beta$. \vs
\item[$\bullet$]
There exists a David map $\varphi:\CC \to \CC$ such that
the Jordan curve $\Gamma=\varphi (\Sen)$ satisfies $\Hdim
\Gamma=2$.\vs
\end{enumerate}
One should contrast the first statement with the fact that
quasiconformal maps preserve sets of Hausdorff dimension $0$ and
$2$. The second statement provides an example of a Jordan curve
with Hausdorff dimension $2$ which is (quasi)conformally
removable.

ims0301

A. Carocca and R. E. Rodr\'\i guez.
Jacobians with group actions and rational idempotents
Abstract: The object of this paper is to prove some general results about
rational idempotents for a finite group $G$ and deduce from them
geometric information about the components that appear in the
decomposition of the Jacobian variety of a curve with $G$action.
We give an algorithm to find explicit primitive rational
idempotents for any $G$, as well as for rational projectors
invariant under any given subgroup. These explicit constructions
allow geometric descriptions of the factors appearing in the
decomposition of a Jacobian with group action: from them we deduce
the decomposition of any Prym or Jacobian variety of an
intermediate cover, in the case of a Jacobian with $G$action. In
particular, we give a necessary and sufficient condition for a
Prym variety of an intermediate cover to be such a factor.

ims0302

Santiago R. Simanca
Heat Flows for Extremal K\"ahler Metrics
Abstract: Let $(M,J,\Omega)$ be a polarized complex manifold of K\"ahler
type. Let $G$ be the maximal compact subgroup of the automorphism
group of $(M,J)$. On the space of K\"ahler metrics that are
invariant under $G$ and represent the cohomology class $\Omega$,
we define a flow equation whose critical points are extremal
metrics, those that minimize the square of the $L^2$norm of the
scalar curvature. We prove that the dynamical system in this
space of metrics defined by the said flow does not have periodic
orbits, and that its only fixed points, or extremal solitons,
are extremal metrics. We prove local time existence of the flow,
and conclude that if the lifespan of the solution is finite, then
the supremum of the norm of its curvature tensor must blowup as
time approaches it. We end up with some conjectures concerning
the plausible existence and convergence of global solutions under
suitable geometric conditions.

ims0401

John W. Milnor
On Latt\`es Maps
Abstract: An exposition of the 1918 paper of Latt\`es and its modern
formulations and applications.

ims0402

R. L. Adler, B. Kitchens, M. Martens, C. Pugh, M. Shub and
Title: Convex Dynamics and Applications
Abstract: This paper proves a theorem about bounding orbits of a time
dependent dynamical system. The maps that are involved are
examples in convex dynamics, by which we mean the dynamics of
piecewise isometries where the pieces are convex. The theorem came
to the attention of the authors in connection with the problem of
digital halftoning. \textit{Digital halftoning} is a family of
printing technologies for getting full color images from only a
few different colors deposited at dots all of the same size. The
simplest version consist in obtaining grey scale images from only
black and white dots. A corollary of the theorem is that for
\textit{error diffusion}, one of the methods of digital
halftoning, averages of colors of the printed dots converge to
averages of the colors taken from the same dots of the actual
images. Digital printing is a special case of a much wider class
of scheduling problems to which the theorem applies. Convex
dynamics has roots in classical areas of mathematics such as
symbolic dynamics, Diophantine approximation, and the theory of
uniform distributions.

ims0403

L. Rempe and D. Schleicher
Bifurcations in the Space of Exponential Maps
Abstract: This article investigates the
parameter space of the exponential family $z\mapsto
\exp(z)+\kappa$. We prove that the boundary (in $\C$) of every
hyperbolic component is a Jordan arc, as conjectured by Eremenko
and Lyubich as well as Baker and Rippon, and
that $\infty$ is not accessible through any nonhyperbolic
(``queer'') stable component. The main part of the argument
consists of demonstrating a general ``Squeezing Lemma'', which
controls the structure of parameter space
near infinity. We also prove a second conjecture of Eremenko and
Lyubich concerning bifurcation trees of hyperbolic components.

ims0404

A. Avila and M. Lyubich
Examples of Feigenbaum Julia sets with small Hausdorff dimension
Abstract: We give examples of infinitely renormalizable quadratic
polynomials $F_c: z\mapsto z^2+c$ with stationary combinatorics
whose Julia sets have Hausdorff dimension arbitrary close to 1.
The combinatorics of the renormalization involved is close to the
Chebyshev one.
The argument is based upon a new tool, a ``Recursive Quadratic
Estimate'' for the Poincar\'e series of an infinitely
renormalizable map.

ims0405

A. Avila and M. Lyubich
Hausdorff dimension and conformal measures of Feigenbaum Julia sets
Abstract: We show that contrary to anticipation suggested by the dictionary
between rational maps and Kleinian groups and by the ``hairiness
phenomenon'', there exist many Feigenbaum
Julia sets $J(f)$ whose Hausdorff dimension is strictly smaller
than two.
We also prove that for any Feigenbaum Julia set,
the Poincar\'e critical exponent $\de_\crit$
is equal to the hyperbolic dimension $\HD_\hyp(J(f))$.
Moreover, if $\area J(f)=0$ then $\HD_\hyp (J(f))=\HD(J(f))$.
In the stationary case, the last statement can be reversed:
if $\area J(f)> 0$ then $\HD_\hyp (J(f))< 2$.
We also give a new construction of conformal measures on $J(f)$
that implies that they exist for any $\de\in [\de_\crit, \infty)$,
and analyze their scaling and dissipativity/conservativity
properties.

ims0406

A. A. Pinto and D. Sullivan
Dynamical Systems Applied to Asymptotic Geometry
Abstract: In the paper we discuss two questions about smooth expanding
dynamical systems on the circle. (i) We characterize the sequences
of asymptotic length ratios which occur for systems with H\"older
continuous derivative. The sequence of asymptotic length ratios are
precisely those given by a positive H\"older continuous function $s$
(solenoid function) on the Cantor set $C$ of $2$adic integers
satisfying a functional equation called the matching condition. The
functional equation for the $2$adic integer Cantor set is $$ s
(2x+1)= \frac{s (x)} {s (2x)}
\left( 1+\frac{1}{ s (2x1)}\right)1.
$$ We also present a onetoone correspondence between solenoid
functions and affine classes of $2$adic quasiperiodic tilings of
the real line that are fixed points of the 2amalgamation
operator. (ii) We calculate the precise maximum possible level of
smoothness for a representative of the system, up to
diffeomorphic conjugacy, in terms of the functions $s$ and
$cr(x)=(1+s(x))/(1+(s(x+1))^{1})$. For example, in the Lipschitz
structure on $C$ determined by $s$, the maximum smoothness is
$C^{1+\alpha}$ for $0 < \alpha \le 1$ if, and only if, $s$ is
$\alpha$H\"older continuous. The maximum smoothness is
$C^{2+\alpha}$ for $0 < \alpha \le 1$ if, and only if, $cr$ is
$(1+\alpha)$H\"older. A curious connection with Mostow type
rigidity is provided by the fact that $s$ must be constant if it
is $\alpha$H\"older for $\alpha > 1$.

ims0501

Carlangelo Liverani, Marco Martens
Convergence to equilibrium for intermittent symplectic maps
Abstract: We investigate a class of area preserving nonuniformly hyperbolic
maps of the two torus. First we establish some results on the
regularity of the invariant foliations, then we use this knowledge
to estimate the rate of mixing.

ims0502

Jeremy Kahn, Mikhail Lyubich
The QuasiAdditivity Law in Conformal Geometry
Abstract: We consider a Riemann surface $S$ of finite type containing a family of $N$ disjoint disks $D_i$,
and prove the following QuasiAdditivity Law:
If the total extremal width $\sum \WW(S\sm D_i)$ is big enough (depending on $N$)
then it is comparable with the extremal width $\WW (S,\cup D_i)$
(under a certain ``separation assumption'') .
We also consider a branched covering $f: U\ra V$ of degree $N$ between two disks
that restricts to a map $\La\ra B$ of degree $d$ on some disk $\La \Subset U$.
We derive from the QuasiAdditivity Law that if $\mod(U\sm \La)$ is sufficiently small,
then (under a ``collar assumption'')
the modulus is quasiinvariant under $f$, namely
$\mod(V\sm B)$ is comparable with $d^2 \mod(U\sm \La)$.
This Covering Lemma has important consequences in holomorphic dynamics which will
be addressed in the forthcoming notes.

ims0503

Jeremy Kahn, Mikhail Lyubich
Local connectivity of Julia sets for unicritical polynomials
Abstract: We prove that the Julia set $J(f)$ of
at most finitely renormalizable unicritical polynomial $f:z\mapsto z^d+c$ with all periodic points repelling
is locally connected. (For $d=2$ it was proved by Yoccoz around 1990.)
It follows from a priori bounds in a modified Principle Nest of puzzle pieces.
The proof of a priori bounds makes use of new analytic tools developed in IMS Preprint #2005/02
that give control of moduli of annuli under maps of high degree.

ims0504

Anca Radulescu
The Connected Isentropes Conjecture in a Space of Quartic Polynomials
Abstract: This note is a shortened version of my dissertation thesis, defended at Stony Brook University in December 2004. It illustrates how dynamic complexity of a system evolves under deformations. The objects I considered are
quartic polynomial maps of the interval that are compositions of two logistic maps.
In the parameter space $P^{Q}$ of such maps, I considered the algebraic curves corresponding to the
parameters for which critical orbits are periodic, and I called such curves left and right bones.
Using quasiconformal surgery methods and rigidity I showed that the bones are simple smooth arcs that
join two boundary points. I also analyzed in detail, using kneading theory, how the combinatorics of
the maps evolves along the bones.
The behavior of the topological entropy function of the polynomials in my family is closely related to
the structure of the boneskeleton. The main conclusion of the paper is that the entropy levelsets in
the parameter space that was studied are connected.

ims0505

Artur Avila, Jeremy Kahn, Mikhail Lyubich and Weixiao Shen
Combinatorial rigidity for unicritical polynomials
Abstract: We prove that any unicritical polynomial $f_c:z\mapsto z^d+c$
which is at most finitely renormalizable and has only repelling
periodic points is combinatorially rigid.
It implies that the connectedness locus (the ``Multibrot set'')
is locally connected at the corresponding parameter values.
It generalizes Yoccoz's Theorem for quadratics to the higher degree
case.

ims0506

R. C. Penner and Dragomir Saric
Teichmuller theory of the punctured solenoid
Abstract: The punctured solenoid $\S$ is an initial object for the category
of punctured surfaces with morphisms given by finite covers
branched only over the punctures. The (decorated) Teichm\"uller
space of $\S$ is introduced, studied, and found to be parametrized
by certain coordinates on a fixed triangulation of $\S$.
Furthermore, a point in the decorated Teichm\"uller space induces
a polygonal decomposition of $\S$ giving a combinatorial
description of its decorated Teichm\"uller space itself. This is
used to obtain a nontrivial set of generators of the modular
group of $\S$, which is presumably the main result of this paper.
Moreover, each word in these generators admits a normal form, and the
natural equivalence relation on
normal forms is described. There is furthermore a
nondegenerate modular group invariant two form on the
Teichm\"uller space of $\S$. All of this structure is in perfect analogy with that
of the decorated Teichm\"uller space of a punctured surface of finite type.

ims0507

A. de Carvalho, M. Lyubich, M. Martens
Renormalization in the Henon family, I: universality but nonrigidity
Abstract: In this paper geometric properties of infinitely renormalizable
real H\'enonlike maps $F$ in $\R^2$ are studied. It is shown that the
appropriately defined renormalizations $R^n F$ converge
exponentially to the onedimensional renormalization fixed
point. The convergence to onedimensional systems is at a
superexponential rate controlled by the average Jacobian and a
universal function $a(x)$. It is also shown that the attracting Cantor
set of such a map has Hausdorff dimension less than 1, but
contrary to the onedimensional intuition, it is not rigid,
does not lie on a smooth curve, and generically has unbounded geometry.

ims0601

A. Bonifant, M. Dabija, J. Milnor
Elliptic curves as attractors in P^2, Part 1: dynamics
Abstract: A study of rational maps of the real or complex projective plane of degree
two or more, concentrating
on those which map an elliptic curve onto itself, necessarily by an
expanding map. We describe relatively
simple examples with a rich variety of exotic dynamical behaviors which are
perhaps familiar to the applied dynamics community but not to specialists
in several complex variables. For example, we describe
smooth attractors with riddled or
intermingled attracting basins, and we observe ``blowout'' bifurcations
when the transverse Lyapunov exponent for the invariant
curve changes sign. In the
complex case, the elliptic curve (a topological torus)
can never have a
trapping neighborhood, yet it can have an attracting basin of large measure
(perhaps even of full measure). We also describe examples where there
appear to be Herman rings (that is topological cylinders mapped to themselves
with irrational rotation number) with open attracting basin.
In some cases we provide proofs, but in other cases the discussion
is empirical, based on numerical computation.

ims0602

A. Epstein, V. Markovic, D. Saric
Extremal maps of the universal hyperbolic solenoid
Abstract: We show that the set of points in the Teichmuller space of the
universal hyperbolic solenoid which do not have a Teichmuller
extremal representative is generic (that is, its complement is the set of the first
kind in the sense of Baire). This is in sharp contrast with the Teichmuller space of a
Riemann surface where at least an open, dense subset has
Teichmuller extremal representatives. In addition, we provide a
sufficient criteria for the existence of Teichmuller extremal
representatives in the given homotopy class. These results
indicate that there is an interesting theory of extremal (and
uniquely extremal) quasiconformal mappings on hyperbolic
solenoids.

ims0603

A. A. Pinto, D. A. Rand
Geometric measures for hyperbolic sets on surfaces
Abstract: We present a moduli space for all hyperbolic basic sets of diffeomorphisms
on surfaces that have an invariant measure that is absolutely
continuous with respect to Hausdorff measure.
To do this we introduce two new invariants:
the measure solenoid function and the cocyclegap pair.
We extend the eigenvalue formula of A. N. Livsic and Ja. G. Sinai
for Anosov diffeomorphisms which preserve an absolutely continuous measure to
hyperbolic basic sets on surfaces which possess an invariant measure absolutely
continuous with respect to Hausdorff measure.
We characterise the Lipschitz conjugacy classes
of such hyperbolic systems in a number of ways, for example,
in terms of eigenvalues of periodic points and Gibbs measures.

ims0604

Sylvain Bonnot, R. C. Penner and Dragomir Saric
A presentation for the baseleaf preserving mapping class group of the punctured solenoid
Abstract: We give a presentation for the baseleaf preserving mapping class
group $Mod(\S )$ of the punctured solenoid $\S$. The generators
for our presentation were introduced previously, and several relations
among them were derived. In addition, we
show that $Mod(\S )$ has no nontrivial central elements. Our main
tool is a new complex of triangulations of the disk upon which $Mod(\S )$ acts.

ims0605

Jeremy Kahn
A priori bounds for some infinitely renormalizable quadratics: I. Bounded primitive combinatorics
Abstract: We prove the \emph{a priori} bounds for infinitely renormalizable quadratic polynomials
of bounded primitive type. This implies the local connectivity of the Mandelbrot set at the corresponding points.

ims0606

Jeremy Kahn, Mikhail Lyubich
A priori bounds for some infinitely renormalizable quadratics: II. Decorations
Abstract: A decoration of the Mandelbrot set $M$ (called also a Misiurewicz limb)
is a part of $M$ cut off by two external rays
landing at some tip of a satellite copy of $M$ attached to the main cardioid.
In this paper we consider infinitely
renormalizable quadratic polynomials satisfying the decoration condition,
which means that the combinatorics of the renormalization operators involved is selected from
a finite family of decorations.
For this class of maps we prove {\it a priori} bounds.
They imply local connectivity of the corresponding Julia sets
and the Mandelbrot set at the corresponding parameter values.

ims0607

Araceli Bonifant, John Milnor
Schwarzian Derivatives and Cylinder Maps
Abstract: We describe the way in which the sign of the Schwarzian
derivative for a family of diffeomorphisms of the interval $I$ affects the
dynamics of an associated manytoone
skew product map of the cylinder $(\R/\Z)\times I$.

ims0701

Vladlen Timorin
The external boundary of the bifurcation locus~$M_2$
Abstract: Consider a quadratic rational selfmap of the Riemann sphere such that one critical point is periodic of period 2, and the other critical point lies on the boundary of its immediate basin of attraction. We will give explicit topological models for all such maps. We also discuss the corresponding parameter picture.

ims0702

V.V.M.S. Chandramouli, M. Martens, W. De Melo, C.P. Tresser
Chaotic Period Doubling
Abstract: The period doubling renormalization operator was introduced by M. Feigenbaum and by P. Coullet and C.
Tresser in the nineteenseventieth to study the asymptotic small
scale geometry of the attractor of onedimensional systems which are
at the transition from simple to chaotic dynamics. This geometry
turns out to not depend on the choice of the map under rather mild
smoothness conditions. The existence of a unique renormalization
fixed point which is also hyperbolic among generic smooth enough
maps plays a crucial role in the corresponding renormalization
theory. The uniqueness and hyperbolicity of the renormalization
fixed point were first shown in the holomorphic context, by means that
generalize to other renormalization operators. It was
then proved that in the space of $C^{2+\alpha}$ unimodal maps,
for $\alpha$ close to one, the period doubling renormalization fixed
point is hyperbolic as well. In this paper we study what happens when one
approaches from below the minimal smoothness thresholds for the
uniqueness and for the hyperbolicity of the period doubling
renormalization generic fixed point. Indeed, our main results
states that in the space of $C^2$ unimodal maps the analytic fixed
point is not hyperbolic and that the same remains true when adding
enough smoothness to get a priori bounds. In this smoother class,
called $C^{2+\cdot}$ the failure of hyperbolicity is tamer than in
$C^2$. Things get much
worse with just a bit less of smoothness than $C^2$ as then even the
uniqueness is lost and other asymptotic behavior become possible. We show
that the period doubling renormalization operator acting on the space of
$C^{1+Lip}$ unimodal maps has infinite topological entropy.

ims0703

Pierre Berger
Persistence of stratification of normally expanded laminations
Abstract: This manuscript complements the HirschPughShub (HPS) theory on persistence of normally hyperbolic laminations and the theorem of Robinson on the structural stability of diffeomorphisms that satisfy Axiom A and the strong transversality condition (SA).
We generalize these results by introducing a geometric object: the stratification of laminations. It is a stratification whose strata are laminations. Our main theorem implies the persistence of some stratifications whose strata are normally expanded. The dynamics is a $C^r$endomorphism of a manifold (which is possibly not invertible). The persistence means that for any $C^r$perturbation of the dynamics, there exists a close $C^r$stratification preserved by the perturbation.
This theorem in its elementary statement (the stratification is constituted by a unique stratum) gives the persistence of normally expanded laminations by endomorphisms, generalizing HPS theory. Another application of this theorem is the persistence, as stratifications, of submanifolds with boundary or corners normally expanded.
Moreover, we remark that SA diffeomorphism gives a canonical stratifications: the stratification whose strata are the stable sets of basic pieces of the spectral decomposition. Our Main theorem then implies the persistence of some ``normally SA'' laminations which are not normally hyperbolic.

ims0704

Jeremy Kahn and Mikhail Lyubich
A priori bounds for some infinitely renormalizable quadratics: III.Molecules.
Abstract: In this paper we prove {\it a priori bounds} for infinitely renormalizable quadratic
polynomials satisfying a ``molecule condition''. Roughly speaking, this condition ensures that
the renormalization combinatorics stay away from the satellite types. These {\it a priori bounds}
imply local connectivity of the corresponding Julia sets and the Mandelbrot set at the corresponding parameter
values.

ims0705

G\"unter Rottenfu{\ss}er, Johannes R\"uckert, Lasse Rempe and Dierk Schleicher
Dynamic rays of boundedtype entire functions
Abstract: We construct an entire function in the EremenkoLyubich class $\B$ whose Julia set has only bounded pathcomponents.
This answers a question of Eremenko from 1989 in the negative.
On the other hand, we show that for many functions in $\B$, in particular those of finite order, every escaping point can be connected to $\infty$ by a curve of escaping points. This gives a partial positive answer to the aforementioned question of Eremenko, and answers a question of Fatou from 1926.

ims0706

Davoud Cheraghi
Combinatorial Rigidity for Some Infinitely Renormalizable Unicritical Polynomials
Abstract: We prove Combinatorial rigidity for infinitely renormalizable unicritical polynomials, $f_c:z \mapsto z^d+c$,
with a priori bounds and some "combinatorial condition".
Combining with \cite{KL2}, this implies local connectivity of the
connectedness locus (the "Mandelbrot set" when $d=2$) at the
corresponding parameter values.

ims0801

Pierre Berger
Persistance des sousvari\'et\'es \`a bord et \`a coins normalement dilat\'ees
Abstract: We show that invariant submanifolds with boundary, and more generally with corners which are normally expanded by an endomorphism are persistent as $a$regular stratifications. This result will be shown in class $C^s$, for $s\ge 1$. We present also a simple example of a submanifold with boundary which is normally expanded but nonpersistent as a differentiable submanifold.

ims0802

Mikhail Lyubich, Marco Martens
Renormalization in the H\'enon family, II: The heteroclinic web
Abstract: We study highly dissipative H\'enon maps
$$
F_{c,b}: (x,y) \mapsto (cx^2by, x)
$$
with zero entropy.
They form a region $\Pi$ in the parameter plane bounded on the left by the curve $W$
of infinitely renormalizable maps.
We prove that MorseSmale maps are dense in $\Pi$,
but there exist infinitely many different topological types
of such maps (even away from $W$).
We also prove that in the infinitely renormalizable case,
the average Jacobian $b_F$ on the attracting Cantor set $\OO_F$ is a topological invariant.
These results come from the analysis of the heteroclinic web
of the saddle periodic points based on the renormalization theory.
Along these lines, we show that the unstable manifolds of the periodic points
form a lamination outside $\OO_F$ if and only if there are no heteroclinic tangencies.

ims0803

Artur Avila, Mikhail Lyubich and Weixiao Shen
Parapuzzle of the Multibrot set\\ and typical dynamics of unimodal maps
Abstract: We study the parameter space of unicritical polynomials $f_c:z\mapsto z^d+c$.
For complex parameters, we prove that for Lebesgue almost every $c$,
the map $f_c$ is either hyperbolic or infinitely renormalizable.
For real parameters, we prove that for Lebesgue almost every $c$,
the map $f_c$ is either hyperbolic, or ColletEckmann, or infinitely renormalizable.
These results are based on controlling the spacing between
consecutive elements in the ``principal nest'' of parapuzzle pieces.

ims0804

Vladlen Timorin
Topological regluing of rational functions
Abstract: Regluing is a topological operation that helps to construct topological models for rational functions on the boundaries of certain hyperbolic components.
It also has a holomorphic interpretation, with the flavor of infinite
dimensional ThurstonTeichm\"uller theory.
We will discuss a topological theory of regluing, and trace a direction
in which a holomorphic theory can develop.

ims0805

Jeremy Kahn, Mikhail Lyubich, and Lasse Rempe
A note on hyperbolic leaves and wild laminations of rational functions
Abstract: We study the affine orbifold laminations that were
constructed in \cite{mishayair}. An important question left open in
\cite{mishayair} is whether these laminations are always locally
compact. We show that this is not the case.
The counterexample we construct has the property that the
\emph{regular leaf space} contains (many) hyperbolic leaves
that intersect the Julia set; whether this can happen is itself
a question raised in \cite{mishayair}.

ims0901

MyongHi Kim, Marco Martens, and Scott Sutherland
A Universal Bound for the Average Cost of Root Finding
Abstract: We analyze a pathlifting algorithm for finding an approximate zero of a
complex polynomial, and show that for any polynomial with distinct roots in the
unit disk, the average number of iterates this algorithm requires is
universally bounded by a constant times the log of the condition number. In
particular, this bound is independent of the degree $d$ of the polynomial. The
average is taken over initial values $z$ with $z = 1 + 1/d$ using uniform
measure.

ims0902

Anna M. Benini
Triviality of fibers for Misiurewicz parameters in the exponential family
Abstract: We consider the family of holomorphic maps $e^z+c$ and show that fibers of postcritically finite parameters are trivial. This can be considered as the first and simplest class of nonescaping parameters for which we can obtain triviality of fibers in the exponential family.

ims0903

Araceli Bonifant, Jan Kiwi, John Milnor
Cubic polynomial maps with periodic critical orbit, Part II: Escape regions
Abstract: The parameter space S_p for monic centered cubic polynomial maps with a
marked critical point of period p is a smooth affine algebraic curve
whose genus increases rapidly with p. Each S_p consists of a compact
connectedness locus together with finitely many escape regions, each of
which is biholomorphic to a punctured disk and is characterized by an
essentially unique Puiseux series. This note with describe the topology
of S_p, and of its smooth compactification, in terms of these escape
regions. It concludes with a discussion of the real sublocus of S_p.

ims1001

P. E. Hazard
Henonlike maps with arbitrary stationary combinatorics
Abstract: We extend the renormalization operator introduced in [3] from perioddoubling
Henonlike maps to Henonlike maps with arbitrary stationary combinatorics. We
show the renormalisation prodcudure also holds in this case if the maps are
taken to be strongly dissipative. We study infinitely renormalizable
maps F and show they have an invariant Cantor set O on which F acts like a
padic adding machine for some p > 1. We then show, as for the perioddoubling
case in [3], the sequence of renormalisations have a universal form, but the
invariant Cantor set O is nonrigid. We also show O cannot possess a continuous
invariant line field.

ims1002

P. E. Hazard, M. Lyubich, M. Martens
Renormalisable Henonlike Maps and Unbounded Geometry
Abstract: We show that given a one parameter family F_{b} of strongly
dissipative infinitely renormalisable Henonlike maps, parametrised by a
quantity called the 'average Jacobian' b, the set of all parameters b such
that F_{b} has a Cantor set with unbounded geometry has full Lebesgue
measure.

ims1003

Artur Avila, Mikhail Lyubich
The full renormalization horseshoe for unimodel maps of higher degree: exponential contraction along hybrid classes
Abstract: We prove exponential contraction of renormalization along hybrid classes
of infinitely renormalizable unimodel maps (with arbitrary combinatorics),
in any even degree d. We then conclude that orbits of renormalization
are asymptotic to the full renormalization horseshoe, which we construct.
Our argument for exponential contraction is based on a precompactness
property of the renormalization operator ("beau bounds"), which is leveraged
in the abstract analysis of holomorphic iteration. Besides greater generality,
it yields a unified approach to all combinatorics and degrees: there is no
need to account for the varied geometric details of the dynamics, which were
the typical source of contraction in previous restricted proofs.

ims1004

Pavel Bleher, Mikhail Lyubich, Roland Roeder
LeeYang zeros for DHL and 2D rational dynamics, I. Foliation of the physical cylinder
Abstract: In a classical work of the 1950's, Lee and Yang proved that the zeros
of the partition functions of a ferromagnetic Ising model always lie
on the unit circle. Distribution of the zeros is physically important
as it controls phase transitions in the model. We study this distribution
for the MigdalKadanoff Diamond Hierarchical Lattice (DHL). In this case,
it can be described in terms of the dynamics of an explicit rational
function R in two variables (the renormalization transformation). We
prove that R is partially hyperbolic on an invariant cylinder C. The
LeeYang zeros are organized in a transverse measure for the centralstable
foliation of RC. Their distribution is absolutely continuous. Its density
is C^infty (and nonvanishing) below the critical temperature. Above the
critical temperature, it is C^infty on an open dense subset, but it vanishes
on the complementary Cantor set of positive measure. This seems to be the
first occasion of a complete rigorous description of the LeeYang
distributions beyond 1D models.

ims1101

Tanya Firsova
The critical locus for complex Henon maps
Abstract: We give a topological model of the critical locus for complex Henon maps
that are perturbations of the quadratic polynomial with disconnected
Julia set.

ims1102

M. Lyubich, M. Martens
Probabilistic universality in twodimensional dynamics
Abstract: In this paper we continue to explore infinitely renormalizable Hénon maps with
small Jacobian. It was shown in [CLM] that contrary to the onedimensional
intuition, the Cantor attractor of such a map is nonrigid and the conjugacy
with the onedimensional Cantor attractor is at most 1/2Hölder. Another
formulation of this phenomenon is that the scaling structure of the Hénon
Cantor attractor differs from its onedimensional counterpart. However, in this paper
we prove that the weight assigned by the canonical invariant measure to these
bad spots tends to zero on microscopic scales. This phenomenon is called
Probabilistic Universality. It implies, in particular, that the Hausdorff
dimension of the canonical measure is universal. In this way, universality and
rigidity phenomena of onedimensional dynamics assume a probabilistic nature in
the twodimensional world.

ims1103

Pavel Bleher, Mikhail Lyubich, Roland Roeder
LeeYangFisher zeros for DHL and 2D rational dynamics, II. Global Pluripotential Interpretation
Abstract: In a classical work of the 1950's, Lee and Yang proved that for fixed
nonnegative temperature, the zeros of the partition functions of a
ferromagnetic Ising model always lie on the unit circle in the complex
magnetic field. Zeros of the partition function in complex temperature
were then considered by Fisher, when the magnetic field is set to zero.
Limiting distributions of LeeYang and of Fisher zeros are physically
important as they control phase transitions in the model. One can also
consider the zeros of the partition function simultaneously in both
complex magnetic field and complex temperature. They form an algebraic
curve called the LeeYangFisher (LYF) zeros. In this paper we study
their limiting distribution for the Diamond Hierarchical Lattice (DHL).
In this case, it can be described in terms of the dynamics of an
explicit rational function R in two variables (the
MigdalKadanoff renormalization transformation). We prove that the
LeeYangFisher zeros are equidistributed with respect to a dynamical
(1,1)current in the projective space. The free energy of the lattice
gets interpreted as the pluripotential of this current. We also
describe some of the properties of the Fatou and Julia sets of the
renormalization transformation.

ims1201

Eric Bedford, John Smillie and Tetsuo Ueda
Parabolic Bifurcations in Complex Dimension 2
Abstract: In this paper we consider parabolic bifurcations of families of
diffeomorphisms in two complex dimensions.

ims1202

John Milnor, with an appendix by A. Poirier
Hyperbolic Components
Abstract: Consider polynomial maps f : C > C of degree d >= 2, or more
generally polynomial maps from a finite union of copies of C
to itself. In the space of suitably normalized maps of this type,
the hyperbolic maps form an open set called the hyperbolic locus.
The various connected components of this hyperbolic locus are
called hyperbolic components, and those hyperbolic components
with compact closure (or equivalently those contained in the
"connectedness locus") are called bounded hyperbolic components.
It is shown that each bouned hyperbolic component is a topological
cell containing a unique postcritically finite map called its
center point. For each degree d, the bounded hyperbolic components
can be separated into finitely many distinct types, each of which
is characterized by a suitable reduced mapping scheme S_f. Any
two components with the same reduced mapping scheme are
canonically biholomorphic to each other. There are similar
statements for real polynomial maps, for polynomial maps with
marked critical points, and for rational maps. Appendix A, by
Alfredo Poirier, proves that every reduced mapping scheme can be
represented by some classical hyperbolic component, made up of
polynomial maps of C. This paper is a revised version of [M2],
which was circulated but not published in 1992.

ims1203

John Milnor
Arithmetic of Unicritical Polynomial Maps
Abstract: This note will study complex polynomial maps of degree n >= 2
with only one critical point.

ims1204

Denis Gaidashev, Tomas Johnson, Marco Martens
Rigidity for infinitely renormalizable areapreserving maps
Abstract: Areapreserving maps have been observed to undergo a universal perioddoubling cascade, analogous to the famous FeigenbaumCoulletTresser period doubling cascade in onedimensional dynamics. A renormalization approach has been used by Eckmann, Koch and Wittwer in a computerassisted proof of existence of a conservative renormalization fixed point. Furthermore, it has been shown that {\it infinitely renormalizable maps} in a neighborhood of this fixed point admit invariant Cantor sets on which the dynamics is ``stable''  the Lyapunov exponents vanish on these sets.
Infinite renormalizability exists in several settings in dynamics, for example, in unimodal maps, dissipative H\'enonlike maps, and conservative H\'enonlike maps. All of these types of maps have associated invariant Cantor sets. The unimodal Cantor sets are rigid: the restrictions of the dynamics to the Cantor sets for any two maps are $C^{1+\alpha}$conjugate. Although, strongly dissipative H\'enon maps can be seen as perturbations of unimodal maps, surprisingly the rigidity breaks down. The Cantor attractors of H\'enon maps with different average Jacobians are not smoothly conjugated. It is conjectured that the average Jacobian determines the rigidity class. This conjecture holds when the Jacobian is identically zero, and in this paper we prove that the conjecture also holds for conservative maps close to the conservative renormalization fixed point.
Rigidity is a consequence of an interplay between the decay of geometry and the convergence rate of renormalization towards
the fixed point. Therefore, to demonstrate rigidity, we prove that the upper bound on the spectral radius of the action of the renormalization derivative on infinitely renormalizable maps is sufficiently small.

ims1205

Marco Martens, Björn Winckler
On the Hyperbolicity of Lorenz Renormalization
Abstract: We consider infinitely renormalizable Lorenz maps with real critical exponent
$\alpha>1$ and combinatorial type which is monotone and satisfies a long
return condition. For these combinatorial types we prove the existence of
periodic points of the renormalization operator, and that each map in the
limit set of renormalization has an associated unstable manifold. An
unstable manifold defines a family of Lorenz maps and we prove that each
infinitely renormalizable combinatorial type (satisfying the above
conditions) has a unique representative within such a family. We also prove
that each infinitely renormalizable map has no wandering intervals and that
the closure of the forward orbits of its critical values is a Cantor
attractor of measure zero.

ims1206

Anna Miriam Benini, Mikhail Lyubich
Repelling periodic points and landing of rays for postsingularly bounded exponential maps
Abstract: We show that repelling periodic points are landing points of
periodic rays for exponential maps whose singular value has bounded orbit.
For polynomials with connected Julia sets, this is a celebrated theorem by
Douady, for which we present a new proof. In both cases we also show that
points in hyperbolic sets are accessible by at least one and at most
finitely many rays. For exponentials this allows us to conclude that the
singular value itself is accessible.

ims1207

Mikhail Lyubich, Han Peters
Classification of invariant Fatou components for dissipative Henon maps
Abstract: Fatou components for rational functions in the Riemann sphere are very well understood and play an important role in our understanding of onedimensional dynamics. In higher dimensions the situation is less well understood. In this work we give a classification of invariant Fatou components for moderately dissipative Henon maps. Most of our methods apply in a much more general setting. In particular we obtain a partial classification of invariant Fatou components for holomorphic endomorphisms of projective space, and we generalize Fatou's Snail Lemma to higher dimensions.

ims1208

Francois Berteloot, Thomas Gauthier
On the geometry of bifurcation currents for quadratic rational maps
Abstract: We describe the behaviour at infinity of the bifurcation current in the moduli space of quadratic rational maps. To this purpose, we extend it to some closed, positive (1,1)current on a twodimensional complex projective space and then compute the Lelong numbers and the selfintersection of the extended current.

ims1301

Romain Dujardin, Mikhail Lyubich
Stability and bifurcations of dissipative polynomial automorphisms of C^2
Abstract: We study stability and bifurcations in holomorphic families of polynomial automorphisms of C^2. We say that such a family is weakly stable over some parameter domain if periodic orbits do not bifurcate there. We first show that this defines a meaningful notion of stability, which parallels in many ways the classical notion of Jstability in onedimensional dynamics. In the second part of the paper, we prove that under an assumption of moderate dissipativity, the parameters displaying homoclinic tangencies are dense in the bifurcation locus. This confirms one of Palis' Conjectures in the complex setting. The proof relies on the formalism of semiparabolic bifurcation and the construction of "critical points" in semiparabolic basins (which makes use of the classical DenjoyCarlemanAhlfors and Wiman Theorems).

ims1401

Mario Bonk, Misha Lyubich, Sergei Merenkov
Quasisymmetries of Sierpinski carpet Julia sets
Abstract: We prove that if $\xi$ is a quasisymmetric homeomorphism between Sierpinski carpets that are the Julia sets of postcriticallyfinite rational maps, then $\xi$ is the restriction of a Mobius transformation to the Julia set. This implies that the group of quasisymmetric homeomorphisms of a Sierpinski carpet Julia set of a postcriticallyfinite rational map is finite.

ims1402

Artem Dudko, Michael Yampolsky
Polytime computability of the Feigenbaum Julia set
Abstract: We present the first example of a polytime computable Julia set with a recurrent critical point: we prove that the Julia set of the Feigenbaum map is computable in polynomial time.

ims1403

Tanya Firsova, Mikhail Lyubich
$\lambda$Lemma for families of Riemann surfaces and the critical loci of complex H\'enon map
Abstract: We prove a version of the classical $\lambda$lemma for holomorphic families of Riemann surfaces. We then use it to show that critical loci for complex H\'{e}non maps that are small perturbations of quadratic polynomials with Cantor Julia sets are all quasiconformally equivalent.

ims1404

R. Adler, T. Nowicki, G. Swirszcz, C. Tresser, S. Winograd
Error Diffusion on Simplices: Invariant Regions, Tessellations and Acuteness
Abstract: The error diffusion algorithm can be considered as a time dependent dynamical system that transforms a sequence of inputs into a sequence of inputs. That dynamical system is a time dependent translation acting on a partition of the phase space Aff, a finite dimensional real affine space, into the Voronoi regions of the set C of vertices of some polytope Pol where the inputs all belong.
Given a sequence inp(i) of inputs that are point in Aff, inp(i) gets added to the error vector e(i), the total vector accumulated so far, that belongs to the (Euclidean) vector space mofelling Aff. The sum inp(i)+e(i) is then again in Aff, thus in a well defined element of the partition of Aff that determines in turns one vertex v(i). The point v(i) of Aff is the ith output, and the new error vector to be used next is e(i+1)=inp(i)+e(i)v(i).
The maps e(i)>e(i+1) and inp(i)+e(i)>inp(i+1)+e(i+1) are two form of error diffusion, respectively in the vector space and affine space. Long term behavior of the algorithm can be deduced from the asymptotic properties of invariant sets, especially from the absorbing ones that serve as traps to all orbits. The existence of invariant sets for arbitrary sequence of inputs has been established in full generality, but in such a context, the invariant sets that are shown to exist are arbitrarily large and only few examples of minimal invariant sets can be described. Since the case of constant input (that corresponds to a time independent translation) has its own interest, we study here the invariant set for constant input for special polytopes that contain the ndimensional regular simplices.
In that restricted context of interest in number theory, we study the properties of the minimal absorbing invariant set and prove that typically those sets are bounded fundamental sets for a discrete lattice generated by the simplex and that the
intersections of those sets with the elements of the partition are fundamental sets for specific derived lattices.

ims1405

Remus Radu and Raluca Tanase
A structure theorem for semiparabolic Henon maps
Abstract: Consider the parameter space $\mathcal{P}_{\lambda}\subset \mathbb{C}^{2}$ of complex H\'enon maps
\[ H_{c,a}(x,y)=(x^{2}+c+ay, ax),\ a\neq 0 \] which have a semiparabolic fixed point with one eigenvalue
$\lambda=e^{2\pi i p/q}$. We give a characterization of those H\'enon maps from the curve $\mathcal{P}_{\lambda}$
that are small perturbations of a quadratic polynomial $p$ with a parabolic fixed point of multiplier $\lambda$.
We prove that there is an open disk of parameters in $\mathcal{P}_{\lambda}$ for which the semiparabolic H\'enon
map has connected Julia set $J$ and is structurally stable on $J$ and $J^{+}$. The Julia set $J^{+}$ has a nice
local description: inside a bidisk $\mathbb{D}_{r}\times \mathbb{D}_{r}$ it is a trivial fiber bundle over $J_{p}$,
the Julia set of the polynomial $p$, with fibers biholomorphic to $\mathbb{D}_{r}$. The Julia set $J$ is homeomorphic
to a quotiented solenoid.

ims1501

Peter Hazard, Marco Martens and Charles Tresser
Infinitely Many Moduli of Stability at the Dissipative Boundary of Chaos
Abstract: In the family of areacontracting Henonlike maps with zero topological entropy we show that there are
maps with infinitely many moduli of stability. Thus one cannot find all the possible topological types for nonchaotic
areacontracting Henonlike maps in a family with finitely many parameters. A similar result, but for the chaotic maps
in the family, became part of the folklore a short time after Henon used such maps to produce what was soon conjectured
to be the first nonhyperbolic strange attractors in R^2. Our proof uses recent results about infinitely renormalisable
areacontracting Henonlike maps; it suggests that the number of parameters needed to represent all possible topological
types for areacontracting Henonlike maps whose sets of periods of their periodic orbits are finite (and in particular
are equal to {1, 2, ...,2^{n1}} or an initial segment of this ntuple) increases with the number of periods. In comparison,
among C^kembeddings of the 2disk with k>0, the maximal moduli number for nonchaotic but non areacontracting maps in
the interior of the set of zeroentropy is infinite.

ims1502

Araceli Bonifant, Xavier Buff and John Milnor
Antipode Preserving Cubic Maps: the Fjord Theorem
Abstract: This note will study a family of cubic rational maps which carry antipodal points of the Riemann sphere to antipodal points. We focus particularly on the fjords, which are part of the central hyperbolic component but stretch out to infinity. These serve to decompose the parameter plane into subsets, each of which is characterized by a corresponding rotation number.

ims1503

Matthieu Arfeux
Reading escaping trees from Hubbard trees in Sn
Abstract: We prove that the parameter space of monic centered cubic polynomials
with a critical point of exact period n = 4 is connected. The techniques
developed for this proof work for every n and provide an interesting relation
between escaping trees of DeMarcoMcMullen and Hubbard trees.

ims1504

Pablo Guarino, Marco Martens, and Welington de Melo
Rigidity of critical circle maps
Abstract: We prove that any two C^4 critical circle maps with the same
irrational rotation number and the same odd criticality are conjugate to each
other by a C^1 circle diffeomorphism. The conjugacy is C^{1+\alpha} for Lebesgue
almost every rotation number.

ims1505

Bjorn Winckler and Marco Martens
Physical Measures for Infinitely Renormalizable Lorenz Maps
Abstract: A physical measure on the attractor of a system describes
the statistical behavior of typical orbits. An example occurs
in unimodal dynamics. Namely, all infinitely renormalizable
unimodal maps have a physical measure. For Lorenz dynamics,
even in the simple case of infinitely renormalizable systems,
the existence of physical measures is more delicate.
In this article we construct examples of infinitely renormalizable
Lorenz maps which do not have a physical measure. A priori bounds
on the geometry play a crucial role in (unimodal) dynamics.
There are infinitely renormalizable Lorenz maps which do not
have a priori bounds. This phenomenon is related to the position
of the critical point of the consecutive renormalizations.
The crucial technical ingredient used to obtain these examples
without a physical measure, is the control of the position of
these critical points.

ims1601

Remus Radu and Raluca Tanase
Semiparabolic tools for hyperbolic Henon maps and continuity of Julia sets in C^2
Abstract: We prove some new continuity results for the Julia sets $J$ and $J^{+}$
of the complex H\'enon map $H_{c,a}(x,y)=(x^{2}+c+ay, ax)$, where $a$ and $c$ are complex
parameters. We look at the parameter space of dissipative H\'enon maps which have a fixed
point with one eigenvalue $(1+t)\lambda$, where $\lambda$ is a root of unity and $t$ is
real and small in absolute value. These maps have a semiparabolic fixed point when
$t$ is $0$, and we use the techniques that we have developed in [RT] for the semiparabolic
case to describe nearby perturbations. We show that for small nonzero $t$, the H\'enon map
is hyperbolic and has connected Julia set. We prove that the Julia sets $J$ and $J^{+}$
depend continuously on the parameters as $t\rightarrow 0$, which is a twodimensional analogue
of radial convergence from onedimensional dynamics. Moreover, we prove that this family
of H\'enon maps is stable on $J$ and $J^{+}$ when $t$ is nonnegative.

ims1602

Tanya Firsova, Mikhail Lyubich, Remus Radu, and Raluca Tanase
Hedgehogs for neutral dissipative germs of holomorphic diffeomorphisms of (C^2, 0)
Abstract: We prove the existence of hedgehogs for germs of complex analytic diffeomorphisms
of (C^2, 0) with a semineutral fixed point at the origin, using topological techniques.
This approach also provides an alternative proof of a theorem of P\'erezMarco on the existence of
hedgehogs for germs of univalent holomorphic maps of (C, 0) with a neutral fixed point.

ims1603

Mikhail Lyubich, Remus Radu, and Raluca Tanase
Hedgehogs in higher dimensions and their applications
Abstract: In this paper we study the dynamics of germs of holomorphic diffeomorphisms of
(C^n, 0) with a fixed point at the origin with exactly one neutral eigenvalue.
We prove that the map on any local center manifold of 0 is quasiconformally conjugate to a
holomorphic map and use this to transport results from one complex dimension to higher dimensions.

ims1604

Mikhail Lyubich and Sergei Merenkov
Quasisymmetries of the basilica and the Thompson group
Abstract: We give a description of the group of all quasisymmetric selfmaps of the Julia set
of f(z)=z^21 that have orientation preserving homeomorphic extensions to the whole plane.
More precisely, we prove that this group is the uniform closure of the group generated by the
Thompson group of the unit circle and an inversion. Moreover, this result is quantitative in the
sense that distortions of the approximating maps are uniformly controlled by the distortion of the given map.

ims1701

Marco Martens and Bjorn Winkler
Instability of renormalization
Abstract: In the theory of renormalization for classical dynamical systems, e.g. unimodal maps
and critical circle maps, topological conjugacy classes are stable manifolds of renormalization.
Physically more realistic systems on the other hand may exhibit instability of renormalization within
a topological class. This instability gives rise to new phenomena and opens up directions of inquiry
that go beyond the classical theory. In phase space it leads to the coexistence phenomenon, i.e.
there are systems whose attractor has bounded geometry but which are topologically conjugate to
systems whose attractor has degenerate geometry; in parameter space it causes dimensional
discrepancy, i.e. a topologically full family has too few dimensions to realize all possible
geometric behavior.

ims1702

Misha Lyubich and Han Peters
Structure of partially hyperbolic Hènon maps
Abstract: We consider the structure of substantially dissipative complex Hènon maps admitting a dominated splitting on the Julia set. The dominated splitting assumption corresponds to the onedimensional assumption that there are no critical points on the Julia set. Indeed, we prove the corresponding description of the Fatou set, namely that it consists of only finitely many components, each either attracting or parabolic periodic. In particular there are no rotation domains, and no wandering components. Moreover, we show that $J = J^\star$ and the dynamics on $J$ is hyperbolic away from parabolic cycles.

ims1703

Artem Dudko and Scott Sutherland
On the Lebesgue measure of the Feigenbaum Julia set
Abstract: We show that the Julia set of the Feigenbaum polynomial
has Hausdorff dimension less than 2 (and consequently it has zero Lebesgue
measure).
This solves a longstanding open question.

ims1704

Dzmitry Dudko, Mikhail Lyubich, and Nikita Selinger
Pacman renormalization and selfsimilarity of the Mandelbrot set near Siegel parameters
Abstract: In the 1980s Branner and Douady discovered a surgery relating
various limbs of the Mandelbrot set. We put this surgery in the
framework of "Pacman Renormalization Theory" that combines features of
quadraticlike and Siegel renormalizations. We show that Siegel
renormalization periodic points (constructed by McMullen in the 1990s)
can be promoted to pacman renormalization periodic points. Then we
prove that these periodic points are hyperbolic with onedimensional
unstable manifold. As a consequence, we obtain the scaling laws for the
centers of satellite components of the Mandelbrot set near the
corresponding Siegel parameters.
