Linearization of generalized interval exchange maps

• 成果类型: Article
• 署名作者: Marmi, Stefano; Moussa, Pierre; Yoccoz, Jean-Christophe
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2012.176.3.5
• 发表日期: 2012
• 页码: 1583-1646
• 关键词: area-preserving flows; affine interval; cohomological equation; metric theory; transformations; SPACE; deviation; SURFACES; decay

摘要

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 >= 2. We prove that, amongst Cr+3 deformations of T-0 which are Cr+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.