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We investigate the dynamics of polynomial semigroups (semigroups generated bya family of polynomial maps on the Riemann sphere) and the random dynamics ofpolynomials on the Riemann sphere. Combining the dynamics of semigroups and thefiberwise (random) dynamics, we give a classification of polynomial semigroups$G$ such that $G$ is generated by a compact family $\Gamma $, the planarpostcritical set of $G$ is bounded, and $G$ is (semi-) hyperbolic. In one ofthe 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 nota 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 aJohn domain. Note that this phenomenon does not hold in the usual iteration ofa single polynomial. Moreover, we consider the dynamics of polynomialsemigroups $G$ such that the planar postcritical set of $G$ is bounded and theJulia set is disconnected. Those phenomena of polynomial semigroups and randomdynamics of polynomials that do not occur in the usual dynamics of polynomialsare systematically investigated.

ergodic theory and dynamical systems/ergodic theory and dynamical systems

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