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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.
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.