Local rigidity for periodic generalised interval exchange transformations

• 成果类型: Article
• 署名作者: Ghazouani, Selim
• 署名单位: Universite Paris Saclay
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01051-3
• 发表日期: 2021
• 页码: 467-520
• 关键词: area-preserving flows; cohomological equation; ergodic averages; deviation; SURFACES; DYNAMICS

摘要

In this article we study local rigidity properties of generalised interval exchange maps using renormalisation methods. We study the dynamics of the renormalisation operator R acting on the space of C-3-generalised interval exchange transformations at fixed points (which are standard periodic type IETs). We show that R is hyperbolic and that the number of unstable direction is exactly that predicted by the ergodic theory of IETs and the work of Forni and Marmi-Moussa-Yoccoz. As a consequence we prove that the local C-1-conjugacy class of a periodic interval exchange transformation, with d intervals, whose associated surface has genus g and whose Lyapounoff exponents are all non zero is a codimension g - 1 + d - 1 C-1-submanifold of the space of C-3-generalised interval exchange transformations. This solves a conjecture analogous to that of Marmi-Moussa-Yoccoz, stated for almost all IETs, in the special case of self-similar IETs.