Welcome to the preprint server of the Institute for Mathematical Sciences at Stony Brook University.
The IMS preprints are also available from the mathematics section of the arXiv e-print server, which offers them in several additional formats. To find the IMS preprints at the arXiv, search for Stony Brook IMS in the report name.
PREPRINTS IN THIS SERIES, IN PDF FORMAT.
* Starred papers have appeared in the journal cited.
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Section 1 describes unpublished work by J.-C. Yoccoz on local connectivity of quadratic Julia sets.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Chapter 3 complements the Yoccoz results by describing a family of examples, due to Douady and Hubbard (unpublished), showing that an infinitely renormalizable quadratic polynomial may have non-locally-connected Julia set.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.
We discuss the Sinai method of proving ergodicity of a discontinuous Hamiltonian system with (non-uniform) hyperbolic behavior.
A distortion theory is developed for $S-$unimodal maps. It will be used to get some geometric understanding of invariant Cantor sets. In particular attracting Cantor sets turn out to have Lebesgue measure zero. Furthermore the ergodic behavior of $S-$unimodal maps is classified according to a distortion property, called the Markov-property.
The Milnor problem on one-dimensional attractors is solved for $S-$unimodal maps with a non-degenerate critical point c. It provides us with a complete understanding of the possible limit behavior for Lebesgue almost every point. This theorem follows from a geometric study of the critical set $\omega(c)$ of a "non-renormalizable" map. It is proven that the scaling factors characterizing the geometry of this set go down to 0 at least exponentially. This resolves the problem of the non-linearity control in small scales. The proofs strongly involve ideas from renormalization theory and holomorphic dynamics.
We prove a technical lemma, the $C^{1+\alpha }$-Denjoy-Koebe distortion lemma, estimating the distortion of a long composition of a $C^{1+\alpha }$ one-dimensional mapping $f:M \rightarrowtail M$ with finitely many, non-recurrent, power law critical points. The proof of this lemma combines the ideas of the distortion lemmas of Denjoy and Koebe.
We study geometrically finite one-dimensional mappings. These are a subspace of $C^{1+\alpha }$ one-dimensional mappings with finitely many, critically finite critical points. We study some geometric properties of a mapping in this subspace. We prove that this subspace is closed under quasisymmetrical conjugacy. We also prove that if two mappings in this subspace are topologically conjugate, they are then quasisymmetrically conjugate. We show some examples of geometrically finite one-dimensional mappings.