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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 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 $SL(2, \mathcal{Z})$.
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.
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.
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
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.
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.
It is shown that for non-hyperbolic real quadratic polynomials topological and quasisymmetric conjugacy classes are the same.
By quasiconformal rigidity, each class has only one representative in the quadratic family, which proves that hyperbolic maps are dense.
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 Grötzsch inequality.
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 $C$.