Dynamics of postcritically bounded polynomial semigroups II: fiberwise dynamics and the Julia sets

TL;DR

This paper explores the complex dynamics of polynomial semigroups with bounded postcritical sets, revealing new phenomena in fiberwise and random Julia sets, including their geometric structures and conditions for being Jordan curves or quasicircles.

Contribution

It introduces novel results on fiberwise dynamics, quasicircle structures, and random Julia sets in polynomial semigroups with bounded postcritical sets, using uniform fiberwise quasiconformal surgery.

Findings

Existence of uncountably many disjoint quasicircles in Julia sets

Conditions under which fiberwise Julia sets are Jordan curves but not quasicircles

Random Julia sets are almost surely Jordan curves but not quasicircles

Abstract

We investigate the dynamics of semigroups generated by polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. Moreover, we investigate the associated random dynamics of polynomials. Furthermore, we investigate the fiberwise dynamics of skew products related to polynomial semigroups with bounded planar postcritical set. Using uniform fiberwise quasiconformal surgery on a fiber bundle, we show that if the Julia set of such a semigroup is disconnected, then there exist families of uncountably many mutually disjoint quasicircles with uniform dilatation which are parameterized by the Cantor set, densely inside the Julia set of the semigroup. Moreover, we give a sufficient condition for a fiberwise Julia set $J_{\gamma}$ to satisfy that $J_{\gamma}$ is a Jordan curve but not a quasicircle, the unbounded component of the complement of $J_{\gamma}$ is a John domain and the bounded component of the complement of $J_{\gamma}$ is not a John domain. We show that under certain conditions, a random Julia set is almost surely a Jordan curve, but not a quasicircle. Many new phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are found and systematically investigated.