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arXiv · 2605.05823

Blaschke-type models for multimodal circle maps

Edson de Faria·Welington de Melo·Pedro A. S. Salomão·Edson Vargas
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Abstract

For each integer $m \geq 1$, we construct a finite-dimensional family of rational maps, given by Blaschke-type products, whose restriction to the unit circle consists of $2m$-multimodal maps. We show that every post-critically finite $2m$-multimodal circle map satisfying natural dynamical conditions is topologically conjugate to a map in this family. Moreover, we prove that this realization is unique up to rotation: two maps in the family that are topologically conjugate on the circle differ by a rigid rotation. In particular, the family provides a canonical model realizing all post-critically finite combinatorics in this class. The proofs combine a detailed description of the critical geometry of these Blaschke-type maps with a Thurston-type fixed point argument for a pull-back operator on the parameter space.

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Edson de Faria, Welington de Melo, Pedro A. S. Salomão, Edson Vargas. 2026-05-07. Blaschke-type models for multimodal circle maps. https://arxiv.org/abs/2605.05823

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