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


A. A. Pinto and D. A. Rand
Geometric measures for hyperbolic sets on surfaces
Abstract:

We present a moduli space for all hyperbolic basic sets of diffeomorphisms on surfaces that have an invariant measure that is absolutely continuous with respect to Hausdorff measure. To do this we introduce two new invariants: the measure solenoid function and the 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.

S. Bonnot, R. C. Penner and D. Saric
A presentation for the baseleaf preserving mapping class group of the punctured solenoid
Abstract:

We give a presentation for the baseleaf preserving mapping class group Mod(H) of the punctured solenoid H. The generators for our presentation were introduced previously, and several relations among them were derived. In addition, we show that Mod(H) has no non-trivial central elements. Our main tool is a new complex of triangulations of the disk upon which Mod(H) acts.

J. Kahn
A priori bounds for some infinitely renormalizable quadratics: I. Bounded primitive combinatorics
Abstract:

We prove the 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.

J. Kahn and M. Lyubich
A priori bounds for some infinitely renormalizable quadratics: II. Decorations
Abstract:

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

A. Bonifant and J. Milnor
Schwarzian Derivatives and Cylinder Maps
Abstract:

We describe the way in which the sign of the Schwarzian derivative for a family of diffeomorphisms of the interval I affects the dynamics of an associated many-to-one skew product map of the cylinder (R/Z)×I.

C. Liverani and M. Martens
Convergence to equilibrium for intermittent symplectic maps
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.

J. Kahn and M. Lyubich
The Quasi-Additivity Law in Conformal Geometry
Abstract:

We consider a Riemann surface S of finite type containing a family of N disjoint disks Di, and prove the following Quasi-Additivity Law: If the total extremal width W(SDi) is big enough (depending on N) then it is comparable with the extremal width W(S,Di) (under a certain ``separation assumption'') . We also consider a branched covering f:UV of degree N between two disks that restricts to a map ΛB of degree d on some disk ΛU. We derive from the Quasi-Additivity Law that if mod(UΛ) is sufficiently small, then (under a ``collar assumption'') the modulus is quasi-invariant under f, namely mod(VB) is comparable with d2mod(UΛ). This Covering Lemma has important consequences in holomorphic dynamics which will be addressed in the forthcoming notes.

J. Kahn, M. Lyubich
Local connectivity of Julia sets for unicritical polynomials
Abstract:

We prove that the Julia set J(f) of at most finitely renormalizable unicritical polynomial f:zzd+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.

A. Radulescu
The Connected Isentropes Conjecture in a Space of Quartic Polynomials
Abstract:

This note is a shortened version of my dissertation thesis, defended at Stony Brook University in December 2004. It illustrates how dynamic complexity of a system evolves under deformations. The objects I considered are quartic polynomial maps of the interval that are compositions of two logistic maps. In the parameter space PQ 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.

A. Avila, J. Kahn, M. Lyubich and W. Shen
Combinatorial rigidity for unicritical polynomials
Abstract:

We prove that any unicritical polynomial fc:zzd+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.

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