Rigidity for piece-wise smooth circle maps and certain GIETs
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 GIETs and 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 . Moreover, if and are GIETs with rotation type combinatorial data, generic rotation number and they are break-equivalent as piecewise circle diffeomorphisms, they are actually -conjugated as circle diffeomorphisms. These results generalize the work of K. Cunha and D. Smania \cite{cunha_rigidity_2014} in the case of piecewise circle maps, where the authors prove an analogous result for GIETs with rotation type combinatorial data and bounded rotation number.
Cite
@article{arxiv.2210.14886,
title = {Rigidity for piece-wise smooth circle maps and certain GIETs},
author = {Przemysław Berk and Frank Trujillo},
journal= {arXiv preprint arXiv:2210.14886},
year = {2024}
}