Modul: MAT760 Ergodic Theory and Dynamical Systems Seminar

## e-Seminar: Renormalisation and rigidity of generalised interval exchange maps

Talk by Dr. Selim Ghazouani

**Date:** 06.04.20 **Time:** 13.45 - 14.45 **Room:** Y27H28

A celebrated result of Herman asserts that a circle diffeomorphism whose rotation number satisfies a mild arithmetic condition is smoothly conjugate to its linear model. One can wonder the extent to which the theory of circle diffeomorphisms extends to smooth, non-linear (generalised) interval exchange transformations (GIET). In particular, is a smooth GIET with irrational "rotation number" always smoothly conjugate to its linear model? Building upon the development of Teichmüller dynamics, Forni and later on Marmi-Moussa-Yoccoz brought to light a finite dimensional set of obstructions for this problem. These obstructions are of purely ergodic-theoretic nature. In this talk, I will present a linearisation result establishing that these obstructions are indeed the only ones, in the case where the rotation number satisfies a strong arithmetic condition. If time permits, I will discuss elements of the proof which centres around the dynamics of a renormalisation operator.