2019/07/12 by Selim Ghazouani, Ghazouani, Selim
Computer Science · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Nonlinear Dynamics and Pattern Formation #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1907.05646
openalex publication_date 2019/07/12 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
In this article we study local rigidity properties of generalised interval\nexchange maps using renormalisation methods. We study the dynamics of the\nrenormalisation operator \R acting on the space of\n\C3-generalised interval exchange transformations at fixed points\n(which are standard periodic type IETs). We show that \R is\nhyperbolic and that the number of unstable direction is exactly that predicted\nby the ergodic theory of IETs and the work of Forni and Marmi-Moussa-Yoccoz. As\na consequence we prove that the local \C1-conjugacy class of a\nperiodic interval exchange transformation, with d intervals, whose associated\nsurface has genus g and whose Lyapounoff exponents are all non zero is a\ncodimension g-1 +d-1 \C1-submanifold of the space of\n\C3-generalised interval exchange transformations. This solves a\nparticular case of a conjecture of Marmi-Moussa-Yoccoz.\n