Nonrigidity of piecewise-smooth circle maps

TL;DR

This paper investigates the rigidity properties of piecewise-smooth circle homeomorphisms with break points, showing that under certain conditions, the conjugation between two such maps is a singular function with zero derivative almost everywhere.

Contribution

It establishes that for certain piecewise-smooth circle maps with matching total jumps and bounded type rotation number, the conjugation is necessarily a singular function, revealing nonrigidity in these systems.

Findings

Conjugation is a singular function with zero derivative almost everywhere.

Matching total jumps and bounded type rotation number lead to nonrigidity.

The results extend understanding of smoothness and rigidity in circle homeomorphisms.

Abstract

Let $f_{i},$ $i=1,2$ be piecewise-smooth $C^{1}$ circle homeomorphisms with two break points, $\log Df_{i},$ $i=1,2$ are absolutely continuous on each continuity intervals of $Df_{i}$ and $D\log Df_{i}\in L^{p}$ for some $p>1.$ Suppose, the jump ratios of $f_{1} $ and $f_{2} $ at their break points do not coincide but have the same total jumps (i.e. the product of jump ratios) and identical irrational rotation number of bounded type. Then the conjugation $h$ between $f_{1} $ and $f_{2} $ is a singular function, i.e. it is continuous on $S^1,$ but $Dh(x)=0$ a.e. with respect to Lebesgue measure.