2022/10/26 by Przemysław Berk, Frank Trujillo, Berk, Przemysław +1
Computer Science · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Nonlinear Dynamics and Pattern Formation #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2210.14886
openalex publication_date 2022/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 C3 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 C1. 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 C1-conjugated as circle diffeomorphisms. These results generalize the work of K. Cunha and D. Smania \citecunharigidity2014 in the case of piecewise C3 circle maps, where the authors prove an analogous result for GIETs with rotation type combinatorial data and bounded rotation number.