Puzzle Pieces, Bi-accessibility, and Connectivity of the Julia Set for Generalized Blaschke Products

TL;DR

This paper investigates the dynamics of generalized Blaschke products, revealing how parameter variations affect Julia set connectivity and identifying bi-accessible cycles that influence Fatou component structure.

Contribution

It provides a topological and combinatorial analysis of Julia set connectivity for a family of rational functions, introducing new insights into bi-accessibility and Fatou components.

Findings

Existence of bi-accessible repelling cycles within Arnold tongues.

Complete characterization of Julia set connectivity for the family.

Exclusion of multiply connected Fatou components absent Herman rings.

Abstract

We study the dynamics of a parametric family of rational functions of odd degree, where each function is a generalized Blaschke product that maps the unit circle onto itself. The action of the Blaschke product restricted to the unit circle defines a circle map, and the parameter space of the family exhibits Arnold tongues. As the parameter varies over an Arnold tongue, the action of the circle map changes from a diffeomorphism to a non-injective endomorphisms. Using a combinatorial study of puzzle pieces, we show that for adjacent parameters inside the Arnold tongues, there exist bi-accessible repelling cycles. This topological feature enables us to exclude the presence of multiply connected Fatou components whenever Herman rings are absent. As a result, we obtain a complete characterization of the connectivity of the Julia set for each member of the parametric family.