Blaschke-type models for multimodal circle maps

TL;DR

This paper constructs a family of rational Blaschke-type maps modeling all post-critically finite multimodal circle maps, providing a canonical and unique realization up to rotation.

Contribution

It introduces a finite-dimensional family of models for multimodal circle maps that captures all post-critically finite cases with uniqueness up to rotation.

Findings

Every post-critically finite multimodal circle map is topologically conjugate to a map in the family.

The family provides a canonical model for all such maps.

The realization is unique up to rotation, ensuring a one-to-one correspondence.

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.