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

Abstract

AbstractWe investigate the dynamics of polynomial semigroups (semigroups generated by a family of polynomial maps on the Riemann sphere $\CCI $) 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 Γ, 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 $\gamma \in \Gamma ^{\NN }$, the Julia set Jγ of γ is a Jordan curve but not a quasicircle, the unbounded component of $\CCI {\setminus } J_{\gamma }$ is a John domain, and the bounded component of $\CC {\setminus } J_{\gamma }$ 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.