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

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 Jγ to satisfy that Jγ is a Jordan curve but not a quasicircle, the unbounded component ofĈ \ Jγ is a John domain and the bounded component of C \ Jγ 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. Definition 1.2. For each rational map g :Ĉ →Ĉ, we set CV (g) := {all critical values of g :Ĉ → C}. Moreover, for each polynomial map g :Ĉ →Ĉ, we set CV * (g) := CV (g) \ {∞}. For a rational semigroup G, we set P (G) := g∈G

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.…”

“…Remark 1.18. There are many hyperbolic polynomial semigroups G = α 1 , α 2 ∈ G dis such that for a generic x ∈ Σ 2 , the fiberwise Julia set J x is a Jordan curve but not quasicircle, the unbounded component of C \ J x is a John domain, and the bounded component of C \ J x is not a John domain (see [31,33,30]). See Figure 1.…”

“…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.…”