Rigidity of square-tiled interval exchange transformations

Sébastien Ferenczi; Pascal Hubert

doi:10.3934/jmd.2019006

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2019, Volume 14: 153-177. Doi: 10.3934/jmd.2019006

This volume Previous Article Dilation surfaces and their Veech groups Next Article Tropical dynamics of area-preserving maps

Rigidity of square-tiled interval exchange transformations

Sébastien Ferenczi and
Pascal Hubert

Aix Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, I2M - UMR 7373, 13453 Marseille, France

To the memory of William Veech whose mathematics were a constant source of inspiration for both authors, and who always showed great kindness to the members of the Marseille school, beginning with its founder Gérard Rauzy

Received: March 2017

Revised: April 23, 2018

Published: March 2019

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Abstract

We look at interval exchange transformations defined as first return maps on the set of diagonals of a flow of direction $ \theta $ on a square-tiled surface: using a combinatorial approach, we show that, when the surface has at least one true singularity both the flow and the interval exchange are rigid if and only if $ \tan\theta $ has bounded partial quotients. Moreover, if all vertices of the squares are singularities of the flat metric, and $ \tan\theta $ has bounded partial quotients, the square-tiled interval exchange transformation $ T $ is not of rank one. Finally, for another class of surfaces, those defined by the unfolding of billiards in Veech triangles, we build an uncountable set of rigid directional flows and an uncountable set of rigid interval exchange transformations.

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Mathematics Subject Classification: Primary: 37E05; Secondary: 37B10.

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Figure 1. Square-tiled surface of Example 30. Letters $ a, b, c, d $ describe the sides identifications

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Figure 2. Building an interval exchange associated to the surface in Example 30

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Figure 3. The square-tiled interval exchange we get in Example 30

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Figure 4. Regular octagon

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Figure 5. Interval exchange transformation in the regular octagon

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Figure 6. Trajectories in direction θ run along the cylinder from J in direction θ_n once, unless they are in the subinterval B

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