Rigidity for piece-wise smooth circle maps and certain GIETs
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909112679268352 |
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| author | Berk, Przemysław Trujillo, Frank |
| author_facet | Berk, Przemysław Trujillo, Frank |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_14886 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Rigidity for piece-wise smooth circle maps and certain GIETs Berk, Przemysław Trujillo, Frank Dynamical Systems 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. |
| title | Rigidity for piece-wise smooth circle maps and certain GIETs |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2210.14886 |