Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents

Anton Zorich

Annales de l'institut Fourier (1996)

  • Volume: 46, Issue: 2, page 325-370
  • ISSN: 0373-0956

Abstract

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We construct a map on the space of interval exchange transformations, which generalizes the classical map on the interval, related to continued fraction expansion. This map is based on Rauzy induction, but unlike its relative kown up to now, the map is ergodic with respect to some finite absolutely continuous measure on the space of interval exchange transformations. We present the prescription for calculation of this measure based on technique developed by W. Veech for Rauzy induction.We study Lyapunov exponents related to this map and show that when the number of intervals is m , and the genus of corresponding surface is g , there are m - 2 g Lyapunov exponents, which are equal to zero, while the remaining 2 g ones are distributed into pairs θ i = - θ m - i + 1 . We present an explicit formula for the largest one.

How to cite

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Zorich, Anton. "Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents." Annales de l'institut Fourier 46.2 (1996): 325-370. <http://eudml.org/doc/75181>.

@article{Zorich1996,
abstract = {We construct a map on the space of interval exchange transformations, which generalizes the classical map on the interval, related to continued fraction expansion. This map is based on Rauzy induction, but unlike its relative kown up to now, the map is ergodic with respect to some finite absolutely continuous measure on the space of interval exchange transformations. We present the prescription for calculation of this measure based on technique developed by W. Veech for Rauzy induction.We study Lyapunov exponents related to this map and show that when the number of intervals is $m$, and the genus of corresponding surface is $g$, there are $m-2g$ Lyapunov exponents, which are equal to zero, while the remaining $2g$ ones are distributed into pairs $\theta _i=-\theta _\{m-i+1\}$. We present an explicit formula for the largest one.},
author = {Zorich, Anton},
journal = {Annales de l'institut Fourier},
keywords = {interval exchange transformation; Gauss measure; Rauzy induction; Lyapunov exponents; orientable measured foliation},
language = {eng},
number = {2},
pages = {325-370},
publisher = {Association des Annales de l'Institut Fourier},
title = {Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents},
url = {http://eudml.org/doc/75181},
volume = {46},
year = {1996},
}

TY - JOUR
AU - Zorich, Anton
TI - Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents
JO - Annales de l'institut Fourier
PY - 1996
PB - Association des Annales de l'Institut Fourier
VL - 46
IS - 2
SP - 325
EP - 370
AB - We construct a map on the space of interval exchange transformations, which generalizes the classical map on the interval, related to continued fraction expansion. This map is based on Rauzy induction, but unlike its relative kown up to now, the map is ergodic with respect to some finite absolutely continuous measure on the space of interval exchange transformations. We present the prescription for calculation of this measure based on technique developed by W. Veech for Rauzy induction.We study Lyapunov exponents related to this map and show that when the number of intervals is $m$, and the genus of corresponding surface is $g$, there are $m-2g$ Lyapunov exponents, which are equal to zero, while the remaining $2g$ ones are distributed into pairs $\theta _i=-\theta _{m-i+1}$. We present an explicit formula for the largest one.
LA - eng
KW - interval exchange transformation; Gauss measure; Rauzy induction; Lyapunov exponents; orientable measured foliation
UR - http://eudml.org/doc/75181
ER -

References

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Citations in EuDML Documents

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  1. Luca Marchese, Khinchin type condition for translation surfaces and asymptotic laws for the Teichmüller flow
  2. Carlos Matheus, Jean-Christophe Yoccoz, David Zmiaikou, Homology of origamis with symmetries
  3. Artur Avila, Sébastien Gouëzel, Jean-Christophe Yoccoz, Exponential mixing for the Teichmüller flow
  4. Raphaël Krikorian, Déviations de moyennes ergodiques, flots de Teichmüller et cocycle de Kontsevich-Zorich
  5. Boris Adamczewski, Codages de rotations et phénomènes d'autosimilarité
  6. Sébastien Ferenczi, Luca Q. Zamboni, Eigenvalues and simplicity of interval exchange transformations
  7. Sébastien Ferenczi, A generalization of the self-dual induction to every interval exchange transformation
  8. Anne Broise-Alamichel, Yves Guivarc'h, Exposants caractéristiques de l'algorithme de Jacobi-Perron et de la transformation associée

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