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**PREPRINTS IN THIS SERIES, IN PDF FORMAT.**

* Starred papers have appeared in the journal cited.

*On Removable Sets for Sobolev Spaces in the Plane*

Let $K$ be a compact subset of $\bar{\textbf{C}} ={\textbf{R}}^2$ and let $K^c$ denote its complement. We say $K\in HR$, $K$ is holomorphically removable, if whenever $F:\bar{\textbf{C}} \to\bar{\textbf{C}}$ is a homeomorphism and $F$ is holomorphic off $K$, then $F$ is a Möbius transformation. By composing with a Möbius 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.

*Geometric Finiteness and Uniqueness for Kleinian Groups with Circle Packing Limit Sets*

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 *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.

*The existence of $\sigma-$finite invariant measures, Applications to real 1-dimensional dynamics*

A general construction for $\sigma-$finite absolutely continuous invariant measure will be presented. It will be shown that the local bounded distortion of the Radon-Nykodym derivatives of $f^n_*(\lambda)$ will imply the existence of a $\sigma-$finite invariant measure for the map $f$ which is absolutely continuous with respect to $\lambda$, a measure on the phase space describing the sets of measure zero.

Furthermore we will discuss sufficient conditions for the existence of $\sigma-$finite invariant absolutely continuous measures for real 1-dimensional dynamical systems.

*Scalings in Circle Maps III*

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.

*Hyperbolic Components in Spaces of Polynomial Maps*

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 $\mathcal{p}^{S}$ of suitably normalized maps of this type, the post-critically bounded maps form a compact subset $\mathcal{C}^{S}$ called the connectedness locus, and the hyperbolic maps in $\mathcal{C}^{S}$ form an open set $\mathcal{H}^{S}$ called the hyperbolic connectedness locus. The various connected components $H_\alpha\subset \mathcal{H}^{S}$ 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 $S(f)$. 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.

*The Teichm&uunlaut;ller Space of the Standard Action of $SL(2,Z)$ on $T^2$ is Trivial*

The group $SL(n,{\bf Z})$ acts linearly on $\mathcal{R}^n$, preserving the integer lattice $\mathcal{Z}^{n} \subset \mathcal{R}^{n}$. The induced (left) action on the n-torus $\mathcal{T}^{n} = \mathcal{R}^{n}/\mathcal{Z}^{n}$ will be referred to as the ``standard action''. It has recently been shown that the standard action of $SL(n,\mathcal{Z})$ on $\mathcal{T}^n$, for $n \geq 3$, is both topologically and smoothly rigid. That is, nearby actions in the space of representations of $SL(n,\mathcal{Z})$ into ${\rm Diff}^{+}(\mathcal{T}^{n})$ 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 $\mathcal{Z}$ action on $\mathcal{T}^{n}$ defined by a single hyperbolic element of $SL(n,\mathcal{Z})$ is topologically rigid, an infinite dimensional space of smooth conjugacy classes occur in a neighborhood of the linear action. The standard action of $SL(2, \mathcal{Z})$ on $\mathcal{T}^2$ 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 $SL(2, \mathcal{Z})$ on $\mathcal{T}^2$ by a homeomorphism $h$, then $h$ is smooth. In fact, it will be shown that this rigidity holds for any non-cyclic subgroup of $SL(2, \mathcal{Z})$.

*Dynamics of certain non-conformal semigroups*

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 [1]). 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.

*Cantor Sets in the Line: Scaling Function and the Smoothness of the Shift Map*

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.

*Problems in Holomorphic Dynamics*

This preprint will be published by Springer-Verlag as a chapter of *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 Teichmüller 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

*Polynomial Diffeomorphisms of $C^2$, IV: The Measure of Maximal Entropy and Laminar Currents*

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.