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 Poisson boundary of a group $G$ with a probability measure $\mu$ on it is the space of ergodic components of the time shift in the path space of the associated random walk. Via a generalization of the classical Poisson formula it gives an integral representation of bounded $\mu$-harmonic functions on $G$. In this paper we develop a new method of identifying the Poisson boundary based on entropy estimates for conditional random walks. It leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the Poisson boundary with natural topological boundaries for several classes of groups: word hyperbolic groups and discontinuous groups of isometries of Gromov hyperbolic spaces, groups with infinitely many ends, cocompact lattices in Cartan--Hadamard manifolds, discrete subgroups of semi-simple Lie groups, polycyclic groups, some wreath and semi-direct products including Baumslag--Solitar groups.
Let $f$ be a quadratic polynomial which has an irrationally indifferent fixed point $\alpha$. Let $z$ be a biaccessible point in the Julia set of $f$. Then:
- In the Siegel case, the orbit of $z$ must eventually hit the critical point of $f$.
- In the Cremer case, the orbit of $z$ must eventually hit the fixed point $\alpha$.
Siegel polynomials with biaccessible critical point certainly exist, but in the Cremer case it is possible that biaccessible points can never exist.
As a corollary, we conclude that the set of biaccessible points in the Julia set of a Siegel or Cremer quadratic polynomial has Brolin measure zero.
We prove that the only possible biaccessible points in the Julia set of a Cremer quadratic polynomial are the Cremer fixed point and its preimages. This gives a partial answer to a question posed by C. McMullen on whether such a Julia set can contain any biaccessible point at all.
Let $f:z \mapsto z^2+c$ be a quadratic polynomial whose Julia set $J$ is locally-connected. We prove that the Brolin measure of the set of biaccessible points in $J$ is zero except when $f(z)=z^2-2$ is the Chebyshev quadratic polynomial for which the corresponding measure is one.
We show that the Feigenbaum-Cvitanović equation can be interpreted as a linearizing equation, and the domain of analyticity of the Feigenbaum fixed point of renormalization as a basin of attraction. There is a natural decomposition of this basin which enables to recover a result of local connectivity by Jiang and Hu for the Feigenbaum Julia set.
We prove that any two real-analytic critical circle maps with cubic critical point and the same irrational rotation number of bounded type are $C^{1+\alpha}$ conjugate for some $\alpha>0$.
We prove that two $C^r$ critical circle maps with the same rotation number of bounded type are $C^{1+\alpha}$ conjugate for some $\alpha>0$ provided their successive renormalizations converge together at an exponential rate in the $C^0$ sense. The number $\alpha$ depends only on the rate of convergence. We also give examples of $C^\infty$ critical circle maps with the same rotation number that are not $C^{1+\beta}$ conjugate for any $\beta>0$.
The aim of this work is to describe the equivalence relations in $\mathbb{Q/Z}$ that arise as the rational lamination of polynomials with all cycles repelling. We also describe where in parameter space one can find a polynomial with all cycles repelling and a given rational lamination. At the same time we derive some consequences that this study has regarding the topology of Julia sets.
New insights into the combinatorial structure of the the Mandelbrot set are given by 'Correspondence' and 'Translation' Principles both conjectured and partially proved by E. Lau and D. Schleicher. We provide complete proofs of these principles and discuss results related to them.
Note: The 'Translation' and 'Correspondence' Principles given earlier turned out to be false in the general case. In April 1999, an errata was added to discuss which parts of the two statements are incorrect and which parts remain true.
We give a new proof that all external rays of the Mandelbrot set at rational angles land, and of the relation between the external angle of such a ray and the dynamics at the landing point. Our proof is different from the original one, given by Douady and Hubbard and refined by Lavaurs, in several ways: it replaces analytic arguments by combinatorial ones; it does not use complex analytic dependence of the polynomials with respect to parameters and can thus be made to apply for non-complex analytic parameter spaces; this proof is also technically simpler. Finally, we derive several corollaries about hyperbolic components of the Mandelbrot set.
Along the way, we introduce partitions of dynamical and parameter planes which are of independent interest, and we interpret the Mandelbrot set as a symbolic parameter space of kneading sequences and internal addresses.