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arXiv · 2504.19704

Dynamical decomposition of generalized interval exchange transformations

Charles Fougeron·Sophie Schmidhuber·Corinna Ulcigrai
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Abstract

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 $1980s$.

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Charles Fougeron, Sophie Schmidhuber, Corinna Ulcigrai. 2025-04-28. Dynamical decomposition of generalized interval exchange transformations. https://arxiv.org/abs/2504.19704

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