Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets

Abstract

We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We show that for such a polynomial semigroup, if A and B are two connected components of the Julia set, then one of A and B surrounds the other. From this, it is shown that each connected component of the Fatou set is either simply or doubly connected. Moreover, we show that the Julia set of such a semigroup is uniformly perfect. An upper estimate of the cardinality of the set of all connected components of the Julia set of such a semigroup is given. By using this, we give a criterion for the Julia set to be connected. Moreover, we show that for any n ∈ N ∪ {ℵ0}, there exists a finitely generated polynomial semigroup with bounded planar postcritical set such that the cardinality of the set of all connected components of the Julia set is equal to n. Many new phenomena of polynomial semigroups that do not occur in the usual dynamics of polynomials are found and systematically investigated.

Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups

“…Hence, for any g ∈ G ∈ G, we have that J(g) is connected. We note, however, that the analogous result for polynomial semigroups does not hold, as there are many examples where G ∈ G but J(G) is not connected (see [38,29,30,31,33]). See also [25] for an analysis of the number of connected components of J(G) involving the inverse limit of the spaces of connected components of the realizations of the nerves of finite coverings U of J(G), where U consists of backward images of J(G) under finite word maps in G. In fact, the number of connected components of the Julia set of a finitely generated rational semigroup is deeply related to a new kind of cohomology (called the "interaction cohomology"), which has been introduced by the second author of this paper.…”

“…For research on (semi)hyperbolicity and the Hausdorff dimension of Julia sets of rational semigroups see [20,21,22,23,24,29,30,31,36,33]. Remark 1.7.…”

“…Also, F. Ren, Z. Gong, and W. Zhou studied such semigroups from the perspective of random dynamical systems (see [39,10]). Note that there is a strong connection between the study of dynamics of rational semigroups and that of random complex dynamics (see [29,30,31,32,33]). For example, for a random dynamical system generated by a family of polynomials, we consider the function T ∞ : C → [0, 1] of the probability of tending to ∞.…”

“…We discuss the dynamics of such polynomial semigroups as well as the structure of their Julia sets. For some properties of polynomial semigroups with a bounded finite postcritical set see also [26,27,29,30,31,25,33]. This paper is a continuation of the program initiated in [29,30,31,33].…”

“…For some properties of polynomial semigroups with a bounded finite postcritical set see also [26,27,29,30,31,25,33]. This paper is a continuation of the program initiated in [29,30,31,33]. One may also see the paper [19], which is based on conference talks by the authors, where preliminary (weaker) versions of a portion of the results contained herein were announced and proven.…”