Linearization of generalized interval exchange maps
Abstract
A standard interval exchange map is a onetoone map of the interval which is locally a translation except at finitely many singularities. We define for such maps, in terms of the RauzyVeech 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 $T_0$ which are $C^{r+3}$ tangent to $T_0$ at the singularities, those which are conjugated to $T_0$ by a $C^r$ diffeomorphism close to the identity form a $C^1$ submanifold of codimension $(g1)(2r+1) +s$. 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 $T_0$.
 Publication:

arXiv eprints
 Pub Date:
 March 2010
 arXiv:
 arXiv:1003.1191
 Bibcode:
 2010arXiv1003.1191M
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Complex Variables;
 Mathematics  Number Theory;
 37C15 (Primary) 37E05 (Secondary) 37J40;
 11J70
 EPrint:
 52 pages. This version includes a new section where we explain how to adapt our result to the setting of perturbations of linear flows on translation surfaces