Rigidity for piecewise smooth homeomorphisms on the circle

TL;DR

This paper establishes conditions under which two piecewise smooth circle homeomorphisms are conjugate via a C^1 diffeomorphism, focusing on combinatorics, derivatives, and nonlinearity constraints.

Contribution

It introduces new criteria involving bounded combinatorics and zero mean nonlinearity for C^1 conjugacy of piecewise smooth circle homeomorphisms.

Findings

Conditions for C^1 conjugacy are characterized.

Zero mean nonlinearity is essential for conjugacy.

Bounded combinatorics restrict map complexity.

Abstract

We find conditions for two piecewise C^{2+\nu} homeomorphisms f and g of the circle to be C^1 conjugate. Besides the restrictions on the combinatorics of the maps (we assume that the maps have bounded combinatorics), and necessary conditions on the one-side derivatives of points where f and g are not differentiable, we also assume zero mean nonlinearity for f and g.