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Cascades in the dynamics of affine interval exchange transformations

Part of: Deformations of analytic structures Low-dimensional dynamical systems Low-dimensional topology

Published online by Cambridge University Press:  29 January 2019

ADRIEN BOULANGER ,
CHARLES FOUGERON  and
SELIM GHAZOUANI
ADRIEN BOULANGER
Affiliation:
Institut de Mathématiques de Marseille, Aix-Marseille Université, Campus de Luminy, 13453 Marseille Cedex 13, France
CHARLES FOUGERON
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
SELIM GHAZOUANI
Affiliation:
Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK email [email protected]
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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.

Keywords

low-dimensional dynamicstopological dynamicsTeichmuller dynamics

MSC classification

Primary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 37E35: Flows on surfaces
Secondary: 57M50: Geometric structures on low-dimensional manifolds 32G15: Moduli of Riemann surfaces, Teichmüller theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 8 , August 2020 , pp. 2073 - 2097
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Copyright
© Cambridge University Press, 2019

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