Cascades in the dynamics of affine interval exchange transformations

Boulanger, Adrien; Fougeron, Charles; Ghazouani, Selim

doi:10.1017/etds.2018.141

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Cascades in the dynamics of affine interval exchange transformations

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Fougeron, Charles
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1017/etds.2018.141
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Boulanger, A., Fougeron, C., & Ghazouani, S. (2020). Cascades in the dynamics of affine interval exchange transformations. Ergodic Theory and Dynamical Systems, 40(8), 2073-2097. doi:10.1017/etds.2018.141.

Cite as: https://hdl.handle.net/21.11116/0000-0006-9D67-6

Abstract

We describe in this article the dynamics of a one-parameter family of affine interval exchange transformations. This amounts to studying the directional foliations of a particular dilatation surface introduced in Duryev et al [Affine surfaces and their Veech groups. Preprint, 2016, arXiv:1609.02130], the Disco surface. We show that this family displays various dynamical behaviours: it is generically dynamically trivial but for a Cantor set of parameters the leaves of the foliations accumulate to a (transversely) Cantor set. This study is achieved through analysis of the dynamics of the Veech group of this surface combined with a modified version of Rauzy induction in the context of affine interval exchange transformations.