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.
It has been known for some time that the topological entropy is a nondecreasing function of the parameter in the real quadratic family, which corresponds to the intuitive idea that more nonlinearity induces more complex dynamical behavior. Polynomial families of higher degree depend on several parameters, so that the very question of monotonicity needs to be reformulated. For instance, one can say the entropy is monotone in a multiparameter family if the isentropes, or sets of maps with the same topological entropy, are connected. Here we reduce the problem of the connectivity of the isentropes in the real cubic families to a weak form of the Fatou conjecture on generic hyperbolicity, which was proved to hold true by C. Heckman. We also develop some tools which may prove to be useful in the study of other parameterized families, in particular a general monotonicity result for stunted sawtooth maps: the stunted sawtooth family of a given shape can be understood as a simple family which realizes all the possible combinatorial structures one can expect with a map of this shape on the basis of kneading theory. Roughly speaking, our main result about real cubic families is that they are as monotone as the stunted sawtooth families with the same shapes because of Heckman's result (there are two posible shapes for cubic maps, depending on the behavior at infinity).
Let $F$ be a quadratic rational map of the sphere which has two fixed Siegel disks with bounded type rotation numbers $\theta$ and $\nu$. Using a new degree 3 Blaschke product model for the dynamics of $F$ and an adaptation of complex a priori bounds for renormalization of critical circle maps, we prove that $F$ can be realized as the mating of two Siegel quadratic polynomials with the corresponding rotation numbers $\theta$ and $\nu$.
Brillouin zones were introduced by Brillouin in the thirties to describe quantum mechanical properties of crystals, that is, in a lattice in $\mathbb{R}^n$. They play an important role in solid-state physics. It was shown by Bieberbach that Brillouin zones tile the underlying space and that each zone has the same area. We generalize the notion of Brillouin Zones to apply to an arbitrary discrete set in a proper metric space, and show that analogs of Bieberbach's results hold in this context. We then use these ideas to discuss focusing of geodesics in orbifolds of constant curvature. In the particular case of the Riemann surfaces $\mathbb{H}^2/\Gamma (k)$ (k=2,3, or 5), we explicitly count the number of geodesics of length $t$ that connect the point $i$ to itself
We give an example of a totally disconnected set $E \subset {\mathbb R}^3$ which is not removable for quasiconformal homeomorphisms, i.e., there is a homeomorphism $f$ of ${\mathbb R}^3$ to itself which is quasiconformal off $E$, but not quasiconformal on all of ${\mathbb R}^3$. The set $E$ may be taken with Hausdorff dimension $2$. The construction also gives a non-removable set for locally biLipschitz homeomorphisms.
We demonstrate the existence of a global attractor A with a Cantor set structure for the renormalization of critical circle mappings. The set A is invariant under a generalized renormalization transformation, whose action on A is conjugate to the two-sided shift.
Motivated by the work of Douady, Ghys, Herman and Shishikura on Siegel quadratic polynomials, we study the one-dimensional slice of the cubic polynomials which have a fixed Siegel disk of rotation number $\theta$, with $\theta$ being a given irrational number of Brjuno type. Our main goal is to prove that when $\theta$ is of bounded type, the boundary of the Siegel disk is a quasicircle which contains one or both critical points of the cubic polynomial. We also prove that the locus of all cubics with both critical points on the boundary of their Siegel disk is a Jordan curve, which is in some sense parametrized by the angle between the two critical points. A main tool in the bounded type case is a related space of degree 5 Blaschke products which serve as models for our cubics. Along the way, we prove several results about the connectedness locus of these cubic polynomials.
We study the dynamics of polynomial mappings $f:{\bf C}^k\to{\bf C}^k$ of degree $d\ge2$ that extend continuously to projective space ${\bf P}^k$. Our approach is to study the dynamics near the hyperplane at infinity and then making a descent to $K$ --- the set of points with bounded orbits --- via external rays.
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.