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PREPRINTS IN THIS SERIES, IN PDF FORMAT.
* Starred papers have appeared in the journal cited.


T. Nowicki and S. van Strien
Polynomial Maps with a Julia Set of Positive Measure
Abstract:

In this paper we shall show that there exists $l_0$ such that for each even integer $l \geq l_0$ there exists $c_1 \in \mathbb{R}$ for which the Julia set of $z \mapsto z^l + c_1$ 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.

R. Galeeva, M. Martens, and C. Tresser
Inducing, Slopes, and Conjugacy Classes
Abstract:

We show that the conjugacy class of an eventually expanding continuous piecewise affine interval map is contained in a smooth codimension 1 submanifold of parameter space. In particular conjugacy classes have empty interior. This is based on a study of the relation between induced Markov maps and ergodic theoretical behavior.

C. Bishop and P. Jones
Hausdorff dimension and Kleinian groups
Abstract:

Let $G$ be a non-elementary, finitely generated Kleinian group, $\Lambda(G)$ its limit set and $\Omega(G) = \overline {\mathbb C} \backslash \Lambda(G)$ its set of discontinuity. Let $\delta(G)$ be the critical exponent for the Poincarè 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 $\dim(\partial \Omega)>1$
  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 $\dim(\Lambda)\leq \liminf \dim(\Lambda_n)$
  6. The Minkowski dimension of $\Lambda$ equals the Hausdorff dimension
  7. If $\text{area}(\Lambda)=0$ 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 Poincarè 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.

F. Przytycki
Iterations of Rational Functions: Which Hyperbolic Components Contain Polynomials?
Abstract:

Let $H^d$ be the set of all rational maps of degree $d\ge 2$ on the Riemann sphere which are expanding on Julia set. We prove that if $f\in H^d$ and all or all but one critical points (or values) are in the immediate basin of attraction to an attracting fixed point then there exists a polynomial in the component $H(f)$ of $H^d$ containing $f$. If all critical points are in the immediate basin of attraction to an attracting fixed point or parabolic fixed point then $f$ restricted to Julia set is conjugate to the shift on the 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 $k < d$ there is no polynomial in $H(f)$. Finally we describe a computer experiment joining an exotic example to a Newton's method (for a polynomial) rational function with a 1-parameter family of rational maps.

J. Kwapisz
A Toral Diffeomorphism with a Non-Polygonal Rotation Set
Abstract:

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

A. Epstein, L. Keen, and C. Tresser
The Set of Maps $F_{a,b}:x \mapsto x+ a+{b\over 2\pi} \sin(2\pi x)$ with any Given Rotation Interval is Contractible.
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.

T. Bedford & A. Fisher
Ratio Geometry, Rigidity and the Scenery Process for Hyperbolic Cantor Sets
Abstract:

Given a $C^{1+\gamma}$ hyperbolic Cantor set $C$, we study the sequence $C_{n,x}$ of Cantor subsets which nest down toward a point $x$ in $C$. We show that $C_{n,x}$ is asymptotically equal to an ergodic Cantor set valued process. The values of this process, called limit sets, are indexed by a H\"older continuous 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.

A. Poirier
Coexistence of Critical Orbit Types in Sub-Hyperbolic Polynomial Maps
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 ${\textbf C}$ 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.

Y. Minsky
Extremal Length Estimates and Product Regions in Teichmüller Space
Abstract:

We study the Teichmüller metric on the Teichmüller 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 Teichmüller 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.

T. Kruger, L. D. Pustyl'nikov, and S. E. Troubetzkoy
Acceleration of Bouncing Balls in External Fields
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.

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