Structure of Julia sets of polynomial semigroups with bounded finite postcritical set

Abstract

Let G be a semigroup of complex polynomials (under the operation of composition of functions) such that there exists a bounded set in the plane which contains any finite critical value of any map g ∈ G. We discuss the dynamics of such polynomial semigroups as well the structure of the Julia set of G. In general, the Julia set of such a semigroup G may be disconnected, and each Fatou component of such G is either simply connected or doubly connected. In this paper, we show that for any two distinct Fatou components of certain types (e.g., two doubly connected components of the Fatou set), the boundaries are separated by a Cantor set of quasicircles (with uniform dilatation) inside the Julia set of G. Furthermore, we provide results concerning the (semi-)hyperbolicity of such semigroups as well as discuss various topological consequences of the postcritically boundedness condition.