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


A. Poirier
On the Realization of Fixed Point Portraits (an addendum to Goldberg & Milnor: Fixed Point Portraits)
Abstract:

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

C. Gole
Periodic Orbits for Hamiltonian systems in Cotangent Bundles
Abstract:

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

Peter Jones
On Removable Sets for Sobolev Spaces in the Plane
Abstract:

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.

L. Keen, B. Maskit, and C. Series
Geometric Finiteness and Uniqueness for Kleinian Groups with Circle Packing Limit Sets
Abstract:

In this paper, we assume that $G$ is a finitely generated torsion free 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.

B. Bielefeld (editor)
Conformal Dynamics Problem List
Abstract:

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

A. M. Blokh and M. Yu. Lyubich
Measurable Dynamics of S-Unimodal Maps of the Interval
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.

J. J. P. Veerman and F. M. Tangerman
On Aubry Mather Sets
Abstract:

Let f be a two dimensional area preserving twist map. For each irrational rotation number in a certain (non trivial) interval, there is an 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.

A.E. Eremenko and M. Yu. Lyubich
Dynamical Properties of Some Classes of Entire Functions
Abstract:

The paper is concerned with the dynamics of an entire transcendental function whose inverse has only finitely many singularities. It is proven 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.

J. Milnor
Dynamics in One Complex Variable: Introductory Lectures
Abstract:

These notes study the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. They are based on introductory lectures given at Stony Brook during the Fall Term of 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.

J. Milnor
Remarks on Iterated Cubic Maps
Abstract:

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

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