A surprising discrepancy in the regularity of conjugacies between generalized interval exchange transformations and their inverses at freezing
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911422236065792 |
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| author | Frączek, Krzysztof Kotlewski, Łukasz |
| author_facet | Frączek, Krzysztof Kotlewski, Łukasz |
| contents | Generalized interval exchange transformations (GIETs) are semi-conjugate to interval exchange transformations (IETs) when the Rauzy-Veech combinatorics is $\infty$-complete. When this semi-conjugacy is a homeomorphism, a fundamental problem is to understand the regularity of the conjugacy and its inverse. Contrary to the usual expectation that their Hölder regularities degenerate simultaneously, we exhibit a strongly asymmetric behavior. For self-similar IETs of hyperbolic periodic type and a natural one-parameter central family of affine IET deformations obtained via a freezing (zero-temperature) limit, the conjugacy becomes arbitrarily irregular while its inverse remains uniformly Hölder. Using thermodynamic formalism for renormalization and zero-temperature limits, we obtain sharp asymptotics for Hausdorff dimensions of invariant and conformal measures and for the supremal Hölder exponents of the conjugacy and its inverse. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_04993 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A surprising discrepancy in the regularity of conjugacies between generalized interval exchange transformations and their inverses at freezing Frączek, Krzysztof Kotlewski, Łukasz Dynamical Systems Mathematical Physics 37E05, 37C15, 37A05, 37A40, 37C45, 37C83, 37D35 Generalized interval exchange transformations (GIETs) are semi-conjugate to interval exchange transformations (IETs) when the Rauzy-Veech combinatorics is $\infty$-complete. When this semi-conjugacy is a homeomorphism, a fundamental problem is to understand the regularity of the conjugacy and its inverse. Contrary to the usual expectation that their Hölder regularities degenerate simultaneously, we exhibit a strongly asymmetric behavior. For self-similar IETs of hyperbolic periodic type and a natural one-parameter central family of affine IET deformations obtained via a freezing (zero-temperature) limit, the conjugacy becomes arbitrarily irregular while its inverse remains uniformly Hölder. Using thermodynamic formalism for renormalization and zero-temperature limits, we obtain sharp asymptotics for Hausdorff dimensions of invariant and conformal measures and for the supremal Hölder exponents of the conjugacy and its inverse. |
| title | A surprising discrepancy in the regularity of conjugacies between generalized interval exchange transformations and their inverses at freezing |
| topic | Dynamical Systems Mathematical Physics 37E05, 37C15, 37A05, 37A40, 37C45, 37C83, 37D35 |
| url | https://arxiv.org/abs/2602.04993 |