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

Zorich, Anton

Annales de l'Institut Fourier, Tome 46 (1996) no. 2, pp. 325-370.

Nous construisons une application sur l’espace des échanges d’intervalles qui généralise l’application classique d’intervalle associée au développement en fraction continue. Cette application est fondée sur l’induction de Rauzy, mais à la différence des fonctions similaires connues jusqu’à présent cette application est ergodique par rapport à une mesure finie absolument continue sur l’espace des échanges d’intervalles. Nous présentons la procédure de calcul de cette mesure fondée sur la technique élaborée par W. Veech pour l’induction de Rauzy.

Nous étudions les exposants de Lyapunov définis par cette application. Soit $m$ le nombre d’intervalles, et soit $g$ le genre de la surface correspondante. Nous montrons qu’il y a $m-2g$ exposants de Lyapunov qui sont égaux à zéro, alors que les autres $2g$ exposants sont distribués en tant que ${\theta }_i=-{\theta }_{m-i+1}$. Nous donnons une formule explicite pour l’exposant le plus grand.

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.

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