[Paper Review] Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups

Rich Stankewitz, Hiroki Sumi|arXiv (Cornell University)|Aug 23, 2007

Mathematical Dynamics and Fractals22 references4 citations

TL;DR

This paper investigates the dynamical and structural properties of Julia sets in postcritically bounded polynomial semigroups, where the postcritical set is uniformly bounded. It proves that distinct Fatou components of certain types (e.g., doubly connected) are separated by a Cantor set of quasicircles with uniform dilatation within the Julia set, establishing a deep topological and quasiconformal structure for such semigroups.

ABSTRACT

We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup $G$ of complex polynomials (under the operation of composition of functions) is such that there exists a bounded set in the plane which contains any finite critical value of any map $g \in G$. 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 (\cite{Su01,Su9}). 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 $G.$ Important in this theory is the understanding of various situations which can and cannot occur with respect to how the Julia sets of the maps $g \in G$ are distributed within the Julia set of the entire semigroup $G$. We give several results in this direction and show how such results are used to generate (semi) hyperbolic semigroups possessing this postcritically boundedness condition.

Motivation & Objective

• To understand the dynamical and structural properties of Julia sets in polynomial semigroups where the postcritical set is bounded.
• To analyze how the Julia sets of individual maps in the semigroup are distributed within the global Julia set of the semigroup.
• To characterize the topological separation between distinct Fatou components, especially doubly connected ones.
• To establish conditions under which semigroups are (semi) hyperbolic under the postcritically boundedness condition.
• To explore the role of quasicircles and uniform quasiconformal distortion in the structure of the Julia set.

Proposed method

• Utilizes the theory of rational and polynomial semigroups under composition, focusing on backward invariance and normality in the Fatou set.
• Applies the concept of the minimal Julia set $ J_{\min}(G) $, which is backward invariant and contains at least three points.
• Employs quasiconformal mappings and distortion estimates (e.g., Koebe Distortion Theorem) to control the geometry of preimages of small balls.
• Uses the uniform boundedness of the postcritical set to ensure that components of preimages remain uniformly separated.
• Applies Sullivan’s No Wandering Domains Theorem to rule out Siegel disks or parabolic cycles in the Fatou set.
• Applies the distortion Lemma 1.10 from [22] to bound the diameter of preimage components under iterated maps, ensuring uniform control.

Experimental results

Research questions

• RQ1 How are the Julia sets of individual maps in a postcritically bounded polynomial semigroup distributed within the global Julia set of the semigroup?
• RQ2 What topological structure separates distinct Fatou components, particularly doubly connected ones, in such semigroups?
• RQ3 Under what conditions does the Julia set of a polynomial semigroup contain a Cantor set of quasicircles with uniform dilatation?
• RQ4 Can the postcritically boundedness condition be used to guarantee (semi) hyperbolicity of the semigroup?
• RQ5 How does the uniform quasiconformal distortion of preimage components affect the global structure of the Julia set?

Key findings

• For any two distinct doubly connected Fatou components in a postcritically bounded polynomial semigroup, their boundaries are separated by a Cantor set of quasicircles with uniform dilatation inside the Julia set.
• The minimal Julia set $ J_{\min}(G) $ has at least three points, and its backward orbits are dense in the full Julia set $ J(G) $.
• Components of preimages of small balls under maps in the semigroup remain uniformly separated due to the bounded postcritical set and uniform quasiconformal distortion.
• The set $ UH(G) $, the union of attracting domains, is disjoint from $ J_{\min}(G) $, implying that points in $ J_{\min}(G) $ are not in the attracting basin of any map in the semigroup.
• For any $ z \in J_{\min}(G) $, there exists a uniform bound $ N $ on the degree of preimages under iterated maps, implying $ z \in SH_N(G) $, the set of points with bounded valence.
• The global Julia set $ J(G) $ contains no wandering domains, and the dynamics on $ J(G) $ is structurally rigid due to the uniform control on preimage components.