Circle homeomorphisms with breaks with no $C^{2-\nu}$ conjugacy

TL;DR

This paper demonstrates that for circle homeomorphisms with breaks, the conjugacy cannot generally be $C^{2- u}$ smooth, even when the maps are analytic outside the break points, highlighting a fundamental difference in rigidity theory.

Contribution

It proves that conjugacy for circle homeomorphisms with breaks may lack $C^{2- u}$ smoothness despite analytic behavior outside breaks, challenging existing rigidity assumptions.

Findings

Conjugacy may not be $C^{2- u}$ smooth for maps with breaks.

Rigidity theory differs significantly from the smooth diffeomorphism case.

Analyticity outside break points does not guarantee high smoothness of conjugacy.

Abstract

The rigidity theory for circle homeomophisms with breaks was studied intensively in the last 20 years. It was proved that under mild conditions of the Diophantine type on the rotation number any two $C^{2+\alpha}$ smooth circle homeomorphisms with a break point are $C^1$ smoothly conjugate to each other, provided that they have the same rotation number and the same size of the break. In this paper we prove that the conjugacy may not be $C^{2-\nu}$ even if the maps are analytic outside of the break points. This result shows that the rigidity theory for maps with singularities is very different from the linearizable case of circle diffeomorphisms where conjugacy is arbitrarily smooth, or even analytic, for sufficiently smooth diffeomorphisms.