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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 $C$.
This paper investigates the existence of Denjoy minimal sets and, more generally, strictly ergodic sets in the dynamics of iterated homeomorphisms. It is shown that for the full two-shift, the collection of such invariant sets with the weak topology contains topological balls of all finite dimensions. One implication is an analogous result that holds for diffeomorphisms with transverse homoclinic points. It is also shown that the union of Denjoy minimal sets is dense in the two-shift and that the set of unique probability measures supported on these sets is weakly dense in the set of all shift-invariant, Borel probability measures.
This will is an expository description of quadratic rational maps.
- Sections 2 through 6 are concerned with the geometry and topology of such maps.
- Sections 7-10 survey of some topics from the dynamics of quadratic rational maps. There are few proofs.
- Section 9 attempts to explore and picture moduli space by means of complex one-dimensional slices.
- Section 10 describes the theory of real quadratic rational maps.
For convenience in exposition, some technical details have been relegated to appendices:
- Appendix A outlines some classical algebra.
- Appendix B describes the topology of the space of rational maps of degree $d$.
- Appendix C outlines several convenient normal forms for quadratic rational maps, and computes relations between various invariants.
- Appendix D describes some geometry associated with the curves $Per_n(\mu)\subset M$.
- Appendix E describes totally disconnected Julia sets containing no critical points.
- Appendix F, written in collaboration with Tan Lei, describes an example of a connected quadratic Julia set for which no two components of the complement have a common boundary point.
This paper concentrates on optical Hamiltonian systems of $T*\mathbb{T}^n$, i.e. those for which $H_{pp}$ is a positive definite matrix, and their relationship with symplectic twist maps. We present theorems of decomposition by symplectic twist maps and existence of periodic orbits for these systems. The novelty of these results resides in the fact that no explicit asymptotic condition is imposed on the system. We also present a theorem of suspension by Hamiltonian systems for the class of symplectic twist map that emerges in our study. Finally, we extend our results to manifolds of negative curvature.