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

We 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.

Pointwise bounded approximation by polynomials

For a bounded open set U ⊂ ℂ, we denote by H∞(U) the collection of all bounded analytic functions on U. We let X denote bdy (U), the boundary of U, Y denote the polynomial hull of U (the complement of the unbounded component of ℂ / X), and U* denote mt (Y), the interior of Y. We denote the sup norm of a function f: A → ℂ by ∥f∥A:We denote the space of all analytic polynomials by ℂ[z], and we denote the open unit disc by D and the unit circle by S1.