Dynamics of postcritically bounded polynomial semigroups III: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles

Abstract

We investigate the dynamics of polynomial semigroups (semigroups generated by a family of polynomial maps on the Riemann sphere) and the random dynamics of polynomials on the Riemann sphere. Combining the dynamics of semigroups and the fiberwise (random) dynamics, we give a classification of polynomial semigroups $G$ such that $G$ is generated by a compact family $\Gamma$, the planar postcritical set of $G$ is bounded, and $G$ is (semi-) hyperbolic. In one of the classes, we have that for almost every sequence $\gamma \in {\Gamma }^{\mathbb{N}}$, the Julia set $J_{\gamma }$ of $\gamma$ is a Jordan curve but not a quasicircle, the unbounded component of the Fatou set $F_{\gamma }$ of $\gamma$ is a John domain, and the bounded component of $F_{\gamma }$ is not a John domain. Note that this phenomenon does not hold in the usual iteration of a single polynomial. Moreover, we consider the dynamics of polynomial semigroups $G$ such that the planar postcritical set of $G$ is bounded and the Julia set is disconnected. Those phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are systematically investigated.

Published in Ergodic Theory Dynam. Systems (2010), Vol. 30, No. 6, 1869--1902. 29 pages, 1 figure, see also http://www.math.sci.osaka-u.ac.jp/~sumi/

References in corpus (1)

• Random complex dynamics and semigroups of holomorphic maps