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

Hiroki Sumi

Published 2008-11-28, updated 2011-01-19Version 5

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 $\gamma \in \Gamma ^{\Bbb{N}}$, 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.