Linearization of generalized interval exchange maps

TL;DR

This paper studies the smooth deformations of generalized interval exchange maps satisfying a specific Diophantine condition, proving that conjugacy classes form smooth submanifolds with codimension depending on the map's combinatorics.

Contribution

It introduces a restricted Roth type condition for interval exchange maps and characterizes the structure of their conjugacy classes under smooth deformations.

Findings

Conjugacy classes form a $C^1$ submanifold of specified codimension.

Almost all maps satisfy the restricted Roth type condition.

Provides a formula for codimension based on combinatorial data.

Abstract

A standard interval exchange map is a one-to-one map of the interval which 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 $T_0$ 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 $(g-1)(2r+1) +s$. 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 $T_0$.