Linearization of generalized interval exchange maps

Abstract

A standard interval exchange map is a one-to-one map of the interval that is locally a translation except at finitely many singularities. We define for such maps, in terms of the Rauzy-Veech continuous fraction algorithm, a diophantine arithmetical condition called restricted Roth type, which is almost surely satisfied in parameter space. Let $T_0$ be a standard interval exchange map of restricted Roth type, and let $r$ be an integer $\geq 2$. We prove that, amongst $C^{r+3}$ deformations of $T_0$ that are $C^{r+3}$ tangent to $T_0$ at the singularities, those that are conjugated to $T_0$ by a $C^r$-diffeomorphism close to the identity form a $C^1$-submanifold of codimension $(g-1)(2r+1) +s$. Here, $g$ is the genus and $s$ is the number of marked points of the translation surface obtained by suspension of $T_0$. Both $g$ and $s$ can be computed from the combinatorics of $T_0$.

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Authors

Stefano Marmi

Scuola Normale
Superiore
Piazza dei Cavalieri, 7
I-56100 Pisa
Italy

Pierre Moussa

Institut de Physique Théorique
CEA/Saclay
91191 Gif-Sur-Yvette
France

Jean-Christophe Yoccoz

Collège de France
75231 Paris Cedex 05
France