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arXiv · 2604.16750

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

Melida Carranza Trejo·Monica Moreno Rocha
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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.

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Melida Carranza Trejo, Monica Moreno Rocha. 2026-04-17. Puzzle Pieces, Bi-accessibility, and Connectivity of the Julia Set for Generalized Blaschke Products. https://arxiv.org/abs/2604.16750

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