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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.19704 |
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| _version_ | 1866913810436063232 |
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| author | Fougeron, Charles Schmidhuber, Sophie Ulcigrai, Corinna |
| author_facet | Fougeron, Charles Schmidhuber, Sophie Ulcigrai, Corinna |
| contents | We develop a renormalization scheme which extends the classical Rauzy-Veech induction used to study interval exchange tranformations (IETs) and allows to study generalized interval exchange transformations (GIETs) $T: [0,1) \to [0,1)$ with possibly more than one quasiminimal component (i.e. not infinite-complete, or, equivalently, not semi-conjugated to a minimal IET). The renormalization is defined for more general maps that we call interval exchange transformations with gaps (g-GIETs), namely partially defined GIETs which appear naturally as the first return map of $C^r$-flows on two-dimensional manifolds to any transversal segment. We exploit this renormalization scheme to find a decomposition of $[0,1)$ into finite unions of intervals which either contain no recurrent orbits, or contain only recurrent orbits which are closed, or contain a unique quasiminimal. This provides an alternative approach to the decomposition results for foliations and flows on surfaces by Levitt, Gutierrez and Gardiner from the $1980s$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19704 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamical decomposition of generalized interval exchange transformations Fougeron, Charles Schmidhuber, Sophie Ulcigrai, Corinna Dynamical Systems We develop a renormalization scheme which extends the classical Rauzy-Veech induction used to study interval exchange tranformations (IETs) and allows to study generalized interval exchange transformations (GIETs) $T: [0,1) \to [0,1)$ with possibly more than one quasiminimal component (i.e. not infinite-complete, or, equivalently, not semi-conjugated to a minimal IET). The renormalization is defined for more general maps that we call interval exchange transformations with gaps (g-GIETs), namely partially defined GIETs which appear naturally as the first return map of $C^r$-flows on two-dimensional manifolds to any transversal segment. We exploit this renormalization scheme to find a decomposition of $[0,1)$ into finite unions of intervals which either contain no recurrent orbits, or contain only recurrent orbits which are closed, or contain a unique quasiminimal. This provides an alternative approach to the decomposition results for foliations and flows on surfaces by Levitt, Gutierrez and Gardiner from the $1980s$. |
| title | Dynamical decomposition of generalized interval exchange transformations |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2504.19704 |