Rigidity for piece-wise smooth circle maps and certain GIETs

TL;DR

This paper proves a rigidity property for conjugacies of certain generalized interval exchange transformations (GIETs), showing under specific conditions that their renormalizations converge and conjugacies are differentiable.

Contribution

It extends existing rigidity results for circle maps to a broader class of GIETs with generic rotation numbers and boundary conditions.

Findings

Renormalizations of conjugate GIETs converge to each other.

Conjugating maps are shown to be $C^1$ under specified conditions.

Results generalize previous work on piecewise $C^3$ circle maps.

Abstract

The goal of this article is to show a rigidity property of conjugacies of generalized interval exchange transformations (GIETs). More precisely, we show that if two piecewise $C^3$ GIETs $f$ and $g$ of generic rotation number with mean-non-linearity 0 are homeomorphic, boundary-equivalent and their renormalizations approach in an appropriate way the set of affine interval exchange transformations, then their respective renormalizations converge to each other and the conjugating map is $C^1$. Moreover, if $f$ and $g$ are GIETs with rotation type combinatorial data, generic rotation number and they are break-equivalent as piecewise circle diffeomorphisms, they are actually $C^1$-conjugated as circle diffeomorphisms. These results generalize the work of K. Cunha and D. Smania \cite{cunha_rigidity_2014} in the case of piecewise $C^3$ circle maps, where the authors prove an analogous result for GIETs with rotation type combinatorial data and bounded rotation number.