Linearization of generalized interval exchange maps

2010/03/05 by Marmi, Stefano, Moussa, Pierre, Yoccoz, Jean-Christophe · 1 citation
#11J70 #37C15 (Primary) 37E05 (Secondary) 37J40 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1003.1191

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 T₀ be a standard interval exchange map of restricted Roth type, and let r be an integer ≥ 2. We prove that, amongst C^r+3 deformations of T₀ which are C^r+3 tangent to T₀ at the singularities, those which are conjugated to T₀ by a C^r diffeomorphism close to the identity form a C¹ 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 T₀. Both g and s can be computed from the combinatorics of T₀.

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