Real bounds and quasisymmetric rigidity of multicritical circle maps

TL;DR

This paper proves that topological conjugacies between certain multicritical circle maps with the same rotation number are quasisymmetric, using real-variable methods that apply to both integer and non-integer critical exponents.

Contribution

It establishes quasisymmetric rigidity for multicritical circle maps with non-flat critical points using purely real-variable techniques, extending previous results to non-integer exponents.

Findings

Topological conjugacy implies quasisymmetry under given conditions.

The proof applies to non-integer critical exponents.

The method avoids complex-analytic machinery.

Abstract

Let $f, g:S^1\to S^1$ be two $C^3$ critical homeomorphisms of the circle with the same irrational rotation number and the same (finite) number of critical points, all of which are assumed to be non-flat, of power-law type. In this paper we prove that if $h:S^1\to S^1$ is a topological conjugacy between $f$ and $g$ and $h$ maps the critical points of $f$ to the critical points of $g$, then $h$ is quasisymmetric. When the power-law exponents at all critical points are integers, this result is a special case of a general theorem recently proved by T.~Clark and S.~van Strien \cite{CS}. However, unlike the proof given in \cite{CS}, which relies on heavy complex-analytic machinery, our proof uses purely real-variable methods, and is valid for non-integer critical exponents as well. We do not require $h$ to preserve the power-law exponents at corresponding critical points.