Symmetric Rigidity for Circle Endomorphisms with Bounded Geometry

TL;DR

This paper establishes a rigidity result for circle endomorphisms with bounded geometry, showing that the conjugacy is symmetric only when it is the identity, thus revealing a strong structural constraint in circle dynamics.

Contribution

It proves a symmetric rigidity theorem for circle endomorphisms with bounded geometry, extending understanding of conjugacy properties in circle dynamics.

Findings

Conjugacy is symmetric only when it is the identity.

Symmetric rigidity applies to a broad class of circle endomorphisms.

Many existing results follow from this general symmetric rigidity principle.

Abstract

Let $f$ and $g$ be two circle endomorphisms of degree $d\geq 2$ such that each has bounded geometry, preserves the Lebesgue measure, and fixes $1$. Let $h$ fixing $1$ be the topological conjugacy from $f$ to $g$. That is, $h\circ f=g\circ h$. We prove that $h$ is a symmetric circle homeomorphism if and only if $h=Id$. Many other rigidity results in circle dynamics follow from this very general symmetric rigidity result.