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Diffusion for chaotic plane sections of 3-periodic surfaces

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Abstract

We study chaotic plane sections of some particular family of triply periodic surfaces. The question about possible behavior of such sections was posed by S. P. Novikov. We prove some estimations on the diffusion rate of these sections using the connection between Novikov’s problem and systems of isometries—some natural generalization of interval exchange transformations. Using thermodynamical formalism, we construct an invariant measure for systems of isometries of a special class called the Rauzy gasket, and investigate the main properties of the Lyapunov spectrum of the corresponding suspension flow.

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Acknowledgments

We heartily thank A. Zorich for posing the problem and several improvements to the first version of the text. We are very grateful to F. Ledrappier who kindly explained Sarig’s theory to us. We also thank I. Dynnikov and V. Delecroix for many fruitful discussions and C. Matheus for his explanations on the Galois version of the twisting/pinching criterium. We thank C. McMullen for the bottom part of the Fig. 1. We also thank the anonymous referee for many useful suggestions and improvements to the previous version of the paper. A. Avila was partially supported by the ERC Starting Grant “Quasiperiodic”and by the Balzan project of Jacob Palis. P. Hubert was partially supported by the projet ANR GeoDyM and ANR VALET. A. Skripchenko was partially supported by the Fondation Sciences Mathématiques de Paris, Metchnikov scholarship and the Dynasty Foundation.

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Authors and Affiliations

  1. CNRS UMR 7586, Institut de Mathématiques de Jussieu, Paris Rive Gauche, Bâtiment Sophie Germain, Case 7021, 75205, Paris Cedex 13, France

    Artur Avila

  2. IMPA, Estrada Dona Castarino 110, 22460-320, Rio de Janeiro, Brazil

    Artur Avila

  3. Institut de Mathématiques de Marseille, 39, rue F. Joliot-Curie, 13453, Marseille Cedex 20, France

    Pascal Hubert

  4. Faculty of Mathematics, National Research University Higher School of Economics, Vavilova St. 7, 112312, Moscow, Russia

    Alexandra Skripchenko

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  1. Artur Avila
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  2. Pascal Hubert
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  3. Alexandra Skripchenko
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Correspondence to Pascal Hubert.

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Avila, A., Hubert, P. & Skripchenko, A. Diffusion for chaotic plane sections of 3-periodic surfaces. Invent. math. 206, 109–146 (2016). https://doi.org/10.1007/s00222-016-0650-z

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