IMS Preprint Server

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.

Abstract:
In this paper we sum up our results on one-dimensional measurable dynamics reducing them to the S-unimodal case (compare Appendix 2). Let f be an S-unimodal 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 Feigenbaum-like attractors is equal to zero. Some extra topological properties of Cantor attractors are studied.

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 f-invariant 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.

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.

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 1989-90. 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 two-dimensional differential geometry.

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.

Abstract:
We consider the space N of C² 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.

Abstract:
Let f be a flat spot circle map with irrational rotation number. Located at the edges of the flat spot are non-flat 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 non-decreasing truncations of non-invertible bimodal circle maps. It follows that the derivatives grow at a sub-exponential rate.

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 ω-limit 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.

Abstract:
Invertible compositions of one-dimensional maps are studied which are assumed to include maps with non-positive 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 non-negative Schwarzian derivative are almost linear-fractional 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)

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.

Abstract:
We study the analytical continuation in the complex plane of free energy of the Ising model on diamond-like 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 Migdal-Kadanoff renorm-group. 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.

Abstract:
We define a class of real numbers that has full measure and is contained in the set of Roth numbers. We prove the C¹ - part of Herman's theorem: if f is a C³ diffeomorphism of the circle to itself with a rotation number ω in this class, then f is C¹ --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.

Abstract:
I. We give a combinatorial analysis of rational rotation subsets of the circle. These are invariant subsets that have well-defined rational rotation numbers under the standard self-covering maps of S¹. 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)=z²+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.

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.

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.

Abstract:
We prove a technical lemma, the $C^{1+\alpha }$-Denjoy-Koebe distortion lemma, estimating the distortion of a long composition of a $C^{1+\alpha }$ one-dimensional mapping $f:M\mapsto M$ with finitely many, non-recurrent, power law critical points. The proof of this lemma combines the ideas of the distortion lemmas of Denjoy and Koebe.

Abstract:
We study geometrically finite one-dimensional mappings. These are a subspace of $C^{1+\alpha}$ one-dimensional 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 one-dimensional mappings.

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 Ω₀(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 H³/G_μ. More precisely, they can be read off from the geometry of the punctured torus ∂C₀/G_μ, where ∂C₀ is the component of the convex hull boundary facing Ω₀(G_μ). The surface ∂C0 has a natural hyperbolic metric and is pleated along geodesics that project to a geodesic lamination λ on ∂C₀/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.

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.

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.

Abstract:
We construct a family of root-finding algorithms which exploit the branched covering structure of a polynomial of degree $d$ with a path-lifting 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.

Abstract:
We prove that the period doubling operator has an expanding direction at the fixed point. We use the induced operator, a ``Perron-Frobenius 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.

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.

Abstract:
We estimate harmonic scalings in the parameter space of a one-parameter family of critical circle maps. These estimates lead to the conclusion that the Hausdorff dimension of the complement of the frequency-locking 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

Abstract:
In this paper we consider the space of smooth conjugacy classes of an Anosov diffeomorphism of the two-torus. The only 2-manifold that supports an Anosov diffeomorphism is the 2-torus, 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.

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 Douady-Hubbard-Yoccoz rigidity theorem follows this result.

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 measure-theoretical attractor into primitive ones, ergodic decomposition and Hopf decomposition. For maps with negative Schwarzian derivative this was done in the series of papers [BL1-BL5], but the approach to the general smooth case must be different.

Abstract:
We study scaling function geometry. We show the existence of the scaling function of a geometrically finite one-dimensional mapping. This scaling function is discontinuous. We prove that the scaling function and the asymmetries at the critical points of a geometrically finite one-dimensional mapping form a complete set of $C^{1}$-invariants within a topological conjugacy class.

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 Ulam-von Neumann type metric.

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 $2End(X)(n-1)$ 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 $2End(X)(p-1)$, 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 $End(X)+1$.

Abstract:
This paper will study topological, geometrical and measure-theoretical 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 non-degenerate 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.

Abstract:
It is shown that some topological equivalency classes of S-unimodal maps are equal to quasisymmetric conjugacy classes. This includes some infinitely renormalizable polynomials of unbounded type.

Abstract:
Let J be a Cantor repellor of a conformal map f. Provided f is a polynomial-like or R-symmetric, 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 non-polynomial algebraic function generating a Cantor repellor on which above two measures coincide.

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, filled-in 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).

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.

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.

Abstract:
We establish that every formal critical portrait (as defined by Goldberg and Milnor), can be realized by a postcritically finite polynomial.

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.

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 Douady-Hubbard description of their structure.

Abstract:
In this paper, we assume that $G$ is a finitely generated torsion free non-elementary 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.

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 non-wandering set are obtained. Also, a sharp transition is found from degenerate geometry similar to what was found earlier for non-differentiable maps with a flat spot to bounded geometry as in critical maps without a flat spot.

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 post-critically 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 post-critically 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.

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 n-torus $\T^{n} = \R^{n}/\Z^{n}$ 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 non-cyclic subgroup of $SL(2, \Z)$.

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.

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.

Abstract:
This preprint will be published by Springer-Verlag 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 Non-Rational Complex Dynamics A. Eremenko and M. Lyubich: Wandering Domains for Holomorphic Maps 5. Newton's Method Scott Sutherland: Bad Polynomials for Newton's Method

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 non-uniformly hyperbolic dynamical systems.

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.

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 non-renormalizable 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 Branner-Hubbard 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 non-locally-connected Julia set. An Appendix describes needed tools from complex analysis, including the Gr\"otzsch inequality.

Abstract:
The theory of Hubbard trees provides an effective classification of non-linear post-critically 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 post-critically finite and nowhere linear will be characterized by a ``forest'', which is made up out of one tree in each copy of \C.

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 two-shift, 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 two-shift and that the set of unique probability measures supported on these sets is weakly dense in the set of all shift-invariant, Borel probability measures.

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 7--10 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 one-dimensional 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.

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.

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 Markov-property.

Abstract:
The Milnor problem on one-dimensional attractors is solved for S-unimodal maps with a non-degenerate 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 "non-renormalizable" 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 non-linearity control in small scales. The proofs strongly involve ideas from renormalization theory and holomorphic dynamics.

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 $f$.

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.

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 Douady-Eremenko-Levin-Petersen theorem on the accessability of periodic sources.

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.

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.

Abstract:
One of the most striking early results in symplectic topology is Gromov's "Non-Squeezing 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 bi-invariant (pseudo-)norm on the group $\Ham(M)$ of compactly supported Hamiltonian symplectomorphisms of the manifold $M$. The deep fact is that this pseudo-norm 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 Non-Squeezing theorem for an arbitrary manifold $M$. We do not need to restrict to manifolds in which the theory of pseudo-holomorphic 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.

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 (ims93-5).

Abstract:
We prove that non-hyperbolic non-renormalizable 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 S-unimodal mappings combinatorially related to rotations of bounded type. We use real methods based on cross-ratios and Schwarzian derivative complemented by complex-analytic estimates in terms of conformal moduli.

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 non-hyperbolic 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 quasi-hyperbolic 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 quasi-conformal (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 pull-back 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).

Abstract:
This paper investigates dynamics that persist under isotopy in classes of orientation-preserving 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.

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.

Abstract:
According to Sullivan, a space ${\cal E}$ of unimodal maps with the same combinatorics (modulo smooth conjugacy) should be treated as an infinitely-dimensional 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 quasi-symmetrically conjugate. This was proved [Ji] and [JS] for some classes of non-renormalizable maps (when the critical point is not too recurrent). Here we consider a space of non-renormalizable 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.

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.

Abstract:
In this paper we shall show that there exists a polynomial unimodal map f: [0,1] -> [0,1] which is 1) non-renormalizable(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.

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.

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.

Abstract:
Let G be a non-elementary, 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 non-differentiable boundary in the the sense that the (inner) tangent points of \partial Omega have zero 1-dimensional measure. Almost every point (with respect to harmonic measure) is a twist point.

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 one-sided shift space of $d$ symbols. We give exotic examples of maps of an arbitrary degree $d$ with a non-simply 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

Abstract:
We construct a diffeomorphism of the two-dimensional torus which is isotopic to the identity and whose rotation set is not a polygon.

Abstract:
Consider the two-parameter 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.

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 set-valued 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.

Abstract:
We establish necessary and sufficient conditions for the realization of mapping schemata as post-critically finite polynomials, or more generally, as post-critically 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.

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.

Abstract:
We introduce two models, the Fermi-Ulam 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.

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.

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.

Abstract:
Using the new diffeomorphism invariants of Seiberg and Witten, a uniqueness theorem is proved for Einstein metrics on compact quotients of irreducible 4-dimensional symmetric spaces of non-compact type. The proof also yields a Riemannian version of the Miyaoka-Yau inequality.

Abstract:
We consider a geometric property of the closest-points 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 quasi-geodesics in Teichm\"uller space, and to estimates of translation distance for pseudo-Anosov maps.

Abstract:
We prove that for continuous maps on the interval, the existence of an n-cycle, implies the existence of n-1 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.

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.

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?''

Abstract:
We suggest a way to associate to a rational map of the Riemann sphere a three dimensional object called a hyperbolic orbifold 3-lamination. The relation of this object to the map is analogous to the relation of a hyperbolic 3-manifold to a Kleinian group. In order to construct the 3-lamination we analyze the natural extension of a rational map and the complex affine structure on the canonical 2-dimensional leaf space contained in it. In this paper the construction is carried out in full for post-critically finite maps. We show that the corresponding laminations have a compact convex core. As a first application we give a three-dimensional proof of Thurston's rigidity for post-critically finite mappings, via the "lamination extension" of the proofs of the Mostow and Marden rigidity and isomorphism theorems for hyperbolic 3-manifolds. An Ahlfors-type 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.

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.

Abstract:
It is shown that a polynomial with a Cremer periodic point has a non-accessible critical point in its Julia set provided that the Cremer periodic point is approximated by small cycles.

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 length-minimizing.

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 non-squeezing theorem both to more general manifolds and to more general capacities. The manifolds we consider are quasi-cylinders, 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 semi-monotone. (This restriction occurs because of the well-known difficulty with the theory of $J$-holomorphic curves in arbitrary $M$.) However, we can only prove the existence of length-minimizing 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 non-trivial and very short loops in $\Ham^c(M)$. When such length-minimizing paths do exist, we can extend the Bialy--Polterovich 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 Hofer-Zehnder capacity seems to be better adapted to the problem at hand, giving sharper estimates in many situations. Also the capacity-area inequality for split cylinders extends more easily to quasi-cylinders in this case. As applications, we generalise Hofer's estimate of the time for which an autonomous flow is length-minimizing 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.

Abstract:
Irreducible 3-manifolds are divided into Haken manifolds and non-Haken 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 non-Haken 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 3-manifolds, whether they are Haken or not, takes on a new significance.

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.

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.

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 self-similar tiling of the plane by fractal tiles known as Gosper Islands.

Abstract:
Let $M$ be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let $\Lambda(M)$ be the supremum of $\lambda_0(N)$ where $N$ varies over all hyperbolic 3-manifolds 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 $\Lambda(M)=D(M)(2-D(M))$ 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 $M$.

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 Gauss--Bonnet type for affine algebraic varieties. The second proof uses Morse theory.

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.

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.

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 a-priori 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.

Abstract:
This paper gives two results that show that the dynamics of a time-periodic 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 Euler-Lagrange flow that are semiconjugate to the geodesic flow of a hyperbolic metric. These results can be viewed as a generalization of the Aubry-Mather theory of twist maps and the Hedlund-Morse-Gromov theory of minimal geodesics on closed surfaces and hyperbolic manifolds.

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 Euler-Lagrange flows of a large class of time-periodic Lagrangian systems. This class contains all time-periodic 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 two-torus in which this gap actually occurs. The same system also gives us an example of a Euler-Lagrange minimizer which is not a Jacobi minimizer on its energy level.

Abstract:
In this work we will show that the Teichm\"{u}ller distance for all elements of a certain class of generalized polynomial-like maps (the class of off-critically hyperbolic generalized polynomial-like 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 polynomial-like 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 non-linear 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 off-crititcally 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 polynomial-like map. The proof follows from the non-existence of invariant affine structure.

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.

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 filled-in Julia set of $z^2-1$. 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.

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 non-hyperbolic and at most finitely renormalizable has a finite absolutely continuous invariant measure.

Abstract:
A sufficient geometrical condition for the existence of absolutely continuous invariant probability measures for S-unimodal maps will be discussed. The Lebesgue typical existence of such measures in the quadratic family will be a consequence.

Abstract:
We prove that a proper holomorphic map on the unit disk in the complex plane is uniquely determined up to post-composition with a Moebius transformation by its critical points.

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

Abstract:
A S.N.Bernstein problem is solved under a natural irreducibility condition. Earlier this result was obtained only in some special case.

Abstract:
We exhibit products of Mandelbrot sets in the two-dimensional complex parameter space of cubic polynomials. These products were observed by J. Milnor in computer experiments which inspired Lavaurs' proof of non local-connectivity 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 two-parameter setting requires further considerations involving ray combinatorics and a pullback argument.

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 finite-dimensional, 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 negative-curvature 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)

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.

Abstract:
In this paper we establish $C^2$ a-priori bounds for the scaling ratios of critical circle mappings in a form that gives also a compactness property for the renormalization operator.

Abstract:
We define a two-sided analog of Erd\"os measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erd\"os measure on the 2-torus. 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 Garsia-Alexander-Zagier and some other results. Besides, we study central measures on the Fibonacci graph, the dynamics of expansions and related questions.

Abstract:
We present a new method to calculate the Hausdorff dimension of a certain class of fractals: boundaries of self-affine 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 lower-bound 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.

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 semi-continuous of the scaling transformation in the $C^0$-topology. With a restriction on the 'non-conformality' (see below) the Hausdorff dimension is lower semi-continous function in the $C^{1}$-topology. We include some examples to show that neither of these notions is continuous.

Abstract:
Pecora and Carroll presented a notion of synchronization where an (n-1)-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 non-synchronizability is stable in the sense that in a full $C^0$ neighborhood of the given map, no homeomorphism is synchronizable.

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 2-dimensional analog of a 1-dimensional universal family.

Abstract:
By considering the way an n-tuple of points in the 2-disk 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.

Abstract:
In this paper we give a combinatorial description of the renormlization limits of infinitely renormalizable unimodal maps with {\it essentially bounded} combinatorics admitting quadratic-like complex extensions. As an application we construct a natural analogue of the period-doubling 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}.

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.

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.

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)

Abstract:
In \cite {HO1}, it was shown that there is a topology on $\C^2\sqcup S^3$ homeomorphic to a 4-ball 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 blow-ups 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.

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.

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 non-complex 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.

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.

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.

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 $\beta>0$.

Abstract:
We prove that any two real-analytic critical circle maps with cubic critical point and the same irrational rotation number of bounded type are $C^{1+\alpha}$ conjugate for some $\alpha>0$.

Abstract:
We show that the Feigenbaum-Cvitanovic 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.

Abstract:
Let $f:z\mapsto z^2+c$ be a quadratic polynomial whose Julia set $J$ is locally-connected. We prove that the Brolin measure of the set of biaccessible points in $J$ is zero except when $f(z)=z^2-2$ is the Chebyshev quadratic polynomial for which the corresponding measure is one.

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

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.

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 Cartan--Hadamard manifolds, discrete subgroups of semi-simple Lie groups, polycyclic groups, some wreath and semi-direct products including Baumslag--Solitar groups.

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.

Abstract:
Motivated by the work of Douady, Ghys, Herman and Shishikura on Siegel quadratic polynomials, we study the one-dimensional 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.

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 two-sided shift.

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 non-removable set for locally biLipschitz homeomorphisms.

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 solid-state 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.

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 $\nu$.

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).

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.

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.

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 filled-in 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 self-contained sequel (IMS Preprint 1998/13a). Finally, we relate triviality of fibers to tuning and renormalization in IMS Preptint 1998/13b.

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.

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 Branner-Douady 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.

Abstract:
We prove a refinement of the Fatou-Shishikura 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 $I-f_*$ on spaces of meromorphic quadratic differentials.

Abstract:
Consider a compact manifold M of dimension at least 2 and the space of $C^r$-smooth diffeomorphisms Diff$^r(M)$. The classical Artin-Mazur 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 A-M property). We extend this result and prove that diffeomorphisms having only hyperbolic periodic points with the A-M property are dense in Diff$^r(M)$. Our proof of this result is much simpler than the original proof of Artin-Mazur.

The second main result is that the A-M 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 Gonchenko-Shilnikov-Turaev, a detailed proof of which is also presented.

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 Douady-Hubbard 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.

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 self-similar about the critical point. We give a lower bound for the ratio of self-similarity, and we show that if $\theta=(\sqrt 5-1)/2$ 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.

Abstract:
We show that the absolute Galois group acts faithfully on the set of Hubbard trees. Hubbard trees are finite planar trees, equipped with self-maps, 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.

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 $\alpha>0$.

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 finite-volume Riemann surfaces -- the simplest nontrivial example that falls into our framework -- and our results on subgroup actions 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 $\R$.

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 two-dimensional 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)

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 orbit-closures for this action are very restricted. Lending support to this conjecture, we show that any orbit-closure 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 number-theoretic result generalizing an isolation theorem of Cassels and Swinnerton-Dyer for products of linear forms. We also obtain similar results for other lattices instead of $SL(n,Z)$, under a suitable irreducibility hypothesis.

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.

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 lower-semi-continuously. 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 co-dimension one, but which is dense. All the other points are conjectured to be points of continuity.

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 non-degenerate case.

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.

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.

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.

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 pseudo-flows in the closure of pseudo-groups generated by perturbations of elements in $\text{Diff}(\Bbb C^n,0)$ on a fixed ball.

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 trans-quasiconformal 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 locally-connected 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.

Abstract:
We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence 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 once-punctured torus, the coefficients coincide with the continued fraction coefficients associated to the ending laminations.

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.

Abstract:
We study dynamics of rational maps that satisfy a decay of geometry condition. Well known conditions of non-uniform 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 quasi-conformal 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.

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.

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 one-dimensional expanding dynamics.

Abstract:
In this paper we extend M.~Lyubich's recent results on the global hyperbolicity of renormalization of quadratic-like germs to the space $\mathbb{U}^r$ of $C^r$ unimodal maps with quadratic critical point. We show that in $\mathbb{U}^r$ the bounded-type 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.

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.

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.

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.

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 $\la=\de(\de-2)$. 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 Anosov--Sinai cocycles, the curvature form, currents and transverse invariant measures, $\la$-harmonic functions, Patterson--Sullivan 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).

Abstract:
This paper gives a proof of the conjectural phenomena on the complex boundary one-dimensional 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 quasi-conformal 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.

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 $0\le \alpha <1$.

Abstract:
Given an orientation-preserving 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 cross-ratio distortion norm $||\cdot ||_{cr}$ of $h$, i.e., there exists a constant $C>0$ such that $$\frac{1}{C}||h||_{cr}\le ||\sigma ||_{Th} \le C||h||_{cr}$$ for any $h$. Secondly we introduce two new norms on the cross-ratio 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 cross-ratio 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}.

Abstract:
In this work we deal with non-discrete subgroups of $\dif$, the group of orientation-preserving 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 non-discrete subgroups of $\s$.

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.

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 $T$.

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 non-repelling 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 non-repelling 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 Bruhat-Tits building of $PSL(2, \C_p)$ and the Berkovich space of $\P(\C_p)$.

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 finite-measure dynamical systems.

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 self-similar 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.

Abstract:
In this paper we prove that in any non-trivial 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 codimension-one analytic submanifolds, ``hybrid classes''. This allows us to transfer the regular or stochastic property of the quadratic family to any non-trivial real analytic family.

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.

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 non-recurrent real quadratics as well as biaccessible points of quadratic Julia sets are given.

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].

Abstract:
A homeomorphism of a compact metric space is {\em tight} provided every non-degenerate 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 semi-conjugacy whose fibers carry zero entropy.

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.

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.

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 blow-up as time approaches it. We end up with some conjectures concerning the plausible existence and convergence of global solutions under suitable geometric conditions.

Abstract:
An exposition of the 1918 paper of Latt\`es and its modern formulations 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.

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.

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.

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.

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 $$ s (2x+1)= \frac{s (x)} {s (2x)} \left( 1+\frac{1}{ s (2x-1)}\right)-1. $$ We also present a one-to-one correspondence between solenoid functions and affine classes of $2$-adic quasiperiodic tilings of the real line that are fixed points of the 2-amalgamation 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 $\alpha > 1$.

Abstract:
We investigate a class of area preserving non-uniformly 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.

Abstract:
We consider a Riemann surface $S$ of finite type containing a family of $N$ disjoint disks $D_i$, and prove the following Quasi-Additivity 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 Quasi-Additivity Law that if $\mod(U\sm \La)$ is sufficiently small, then (under a ``collar assumption'') the modulus is quasi-invariant 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.

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.

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 bone-skeleton. The main conclusion of the paper is that the entropy level-sets in the parameter space that was studied are connected.

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.

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 non-trivial 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 non-degenerate 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.

Abstract:
In this paper geometric properties of infinitely renormalizable real H\'enon-like maps $F$ in $\R^2$ are studied. It is shown that the appropriately defined renormalizations $R^n F$ converge exponentially to the one-dimensional renormalization fixed point. The convergence to one-dimensional systems is at a super-exponential 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 one-dimensional intuition, it is not rigid, does not lie on a smooth curve, and generically has unbounded geometry.

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.

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.

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 cocycle-gap 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.

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 non-trivial central elements. Our main tool is a new complex of triangulations of the disk upon which $Mod(\S )$ acts.

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.

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.

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 many-to-one skew product map of the cylinder $(\R/\Z)\times I$.

Abstract:
Consider a quadratic rational self-map 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.

Abstract:
The period doubling renormalization operator was introduced by M. Feigenbaum and by P. Coullet and C. Tresser in the nineteen-seventieth to study the asymptotic small scale geometry of the attractor of one-dimensional 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.

Abstract:
This manuscript complements the Hirsch-Pugh-Shub (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.

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.

Abstract:
We construct an entire function in the Eremenko-Lyubich class $\B$ whose Julia set has only bounded path-components. 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.

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.

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 non-persistent as a differentiable submanifold.

Abstract:
We study highly dissipative H\'enon maps $$ F_{c,b}: (x,y) \mapsto (c-x^2-by, x) $$ 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 Morse-Smale 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.

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 Collet-Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the ``principal nest'' of parapuzzle pieces.

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 Thurston--Teichm\"uller theory. We will discuss a topological theory of regluing, and trace a direction in which a holomorphic theory can develop.

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}.

Abstract:
We analyze a path-lifting 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.

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 non-escaping parameters for which we can obtain triviality of fibers in the exponential family.

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 sub-locus of S_p.

Abstract:
We extend the renormalization operator introduced in [3] from period-doubling Henon-like maps to Henon-like 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 p-adic adding machine for some p > 1. We then show, as for the period-doubling case in [3], the sequence of renormalisations have a universal form, but the invariant Cantor set O is non-rigid. We also show O cannot possess a continuous invariant line field.

Abstract:
We show that given a one parameter family F_b of strongly dissipative infinitely renormalisable Henon-like 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.

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.

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 Migdal-Kadanoff 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 Lee-Yang zeros are organized in a transverse measure for the central-stable foliation of R|C. Their distribution is absolutely continuous. Its density is C^infty (and non-vanishing) 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 Lee-Yang distributions beyond 1D models.

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.

Abstract:
In this paper we continue to explore infinitely renormalizable Hénon maps with small Jacobian. It was shown in [CLM] that contrary to the one-dimensional intuition, the Cantor attractor of such a map is non-rigid and the conjugacy with the one-dimensional Cantor attractor is at most 1/2-Hölder. Another formulation of this phenomenon is that the scaling structure of the Hénon Cantor attractor differs from its one-dimensional 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 one-dimensional dynamics assume a probabilistic nature in the two-dimensional world.

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 Lee-Yang 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 Lee-Yang-Fisher (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 Migdal-Kadanoff renormalization transformation). We prove that the Lee-Yang-Fisher 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.

Abstract:
In this paper we consider parabolic bifurcations of families of diffeomorphisms in two complex dimensions.

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 post-critically 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.

Abstract:
This note will study complex polynomial maps of degree n >= 2 with only one critical point.

Abstract:
Area-preserving maps have been observed to undergo a universal period-doubling cascade, analogous to the famous Feigenbaum-Coullet-Tresser period doubling cascade in one-dimensional dynamics. A renormalization approach has been used by Eckmann, Koch and Wittwer in a computer-assisted 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\'enon-like maps, and conservative H\'enon-like 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.

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.

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.

Abstract:
Fatou components for rational functions in the Riemann sphere are very well understood and play an important role in our understanding of one-dimensional 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.

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 two-dimensional complex projective space and then compute the Lelong numbers and the self-intersection of the extended current.

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 J-stability in one-dimensional 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 semi-parabolic bifurcation and the construction of "critical points" in semiparabolic basins (which makes use of the classical Denjoy-Carleman-Ahlfors and Wiman Theorems).

Abstract:
We prove that if $\xi$ is a quasisymmetric homeomorphism between Sierpinski carpets that are the Julia sets of postcritically-finite 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 postcritically-finite rational map is finite.

Abstract:
We present the first example of a poly-time computable Julia set with a recurrent critical point: we prove that the Julia set of the Feigenbaum map is computable in polynomial time.

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.

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 i-th 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 n-dimensional 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.

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 semi-parabolic 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 semi-parabolic 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.

Abstract:
In the family of area-contracting Henon-like 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 non-chaotic area-contracting Henon-like 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 non-hyperbolic strange attractors in R^2. Our proof uses recent results about infinitely renormalisable area-contracting Henon-like maps; it suggests that the number of parameters needed to represent all possible topological types for area-contracting Henon-like maps whose sets of periods of their periodic orbits are finite (and in particular are equal to {1, 2, ...,2^{n-1}} or an initial segment of this n-tuple) increases with the number of periods. In comparison, among C^k-embeddings of the 2-disk with k>0, the maximal moduli number for non-chaotic but non area-contracting maps in the interior of the set of zero-entropy is infinite.

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.

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 DeMarco-McMullen and Hubbard trees.

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.

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.

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 semi-parabolic fixed point when $t$ is $0$, and we use the techniques that we have developed in [RT] for the semi-parabolic 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 two-dimensional analogue of radial convergence from one-dimensional dynamics. Moreover, we prove that this family of H\'enon maps is stable on $J$ and $J^{+}$ when $t$ is nonnegative.

Abstract:
We prove the existence of hedgehogs for germs of complex analytic diffeomorphisms of (C^2, 0) with a semi-neutral fixed point at the origin, using topological techniques. This approach also provides an alternative proof of a theorem of P\'erez-Marco on the existence of hedgehogs for germs of univalent holomorphic maps of (C, 0) with a neutral fixed point.

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.

Abstract:
We give a description of the group of all quasisymmetric self-maps of the Julia set of f(z)=z^2-1 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.

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.

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 one-dimensional 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.

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 long-standing open question.

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 quadratic-like 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 one-dimensional 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.