Quasisymmetric orbit-flexibility of multicritical circle maps

TL;DR

This paper investigates the geometric equivalence of orbits in multicritical circle maps, revealing uncountably many classes for typical rotation numbers and constructing non-quasisymmetric conjugacies, thus highlighting complex orbit structures.

Contribution

It proves uncountably many orbit classes for typical rotation numbers and constructs abundant non-quasisymmetric conjugacies in multicritical circle maps.

Findings

Uncountably many orbit classes for Lebesgue typical rotation numbers.

Existence of abundant non-quasisymmetric topological conjugacies.

Application of ergodic theory to classify orbit equivalence.

Abstract

Two given orbits of a minimal circle homeomorphism $f$ are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with $f$. By a well-known theorem due to Herman and Yoccoz, if $f$ is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. As it follows from the a-priori bounds of Herman and Swiatek, the same holds if $f$ is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if $f$ is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in $(0,1)$, then the number of equivalence classes is uncountable (Theorem A). The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. As a by-product of our techniques, we construct topological conjugacies between multicritical circle maps which are not quasisymmetric, and we show that this phenomenon is abundant, both from the topological and measure-theoretical viewpoints (Theorems B and C).