Institute of Mathematics


Modul:   MAT760  Ergodic Theory and Dynamical Systems Seminar

Rigidity of Poincare sections of higher genus flows

Talk by Prof. Dr. Corinna Ulcigrai

Date: 28.02.22  Time: 14.05 - 15.05  Room: ETH HG G 43

A celebrated result in the theory circle diffeomorphisms proved by Michael Herman and Jean-Christophe Yoccoz shows that (smooth) circle diffeomorphisms with a Diophantine rotation numbers are smoothly conjugated to their linear model, namely a rotation of the circle with the same rotation number. In terms of foliations on surfaces, this result shows that, under a full measure condition, smooth, orientable minimal foliations on surfaces of genus one are geometrically rigid, i.e. when they are topologically conjugated to a linear foliation, the conjugacy is also differentiable (and actually smooth). We prove a generalization of this result to higher genus, by showing that, under a full measure condition, smooth, orientable minimal foliations with Morse saddles on a surfaces of genus two are differentiably conjugated to their linear model (and actually the conjugacy can be shown to be C^{1+\alpha}). The result can be rephrased in terms of generalized interval exchange maps (GIETs), namely piecewise diffeomorphisms which appear as Poincare' sections, and conjugacy to their linear models, namely (standard) interval exchange maps (IETs) and is proved using renormalization and proving a dynamical dichotomy for the renormalization operator (which is valid in any genus). This result proves a conjecture on GIETs by Marmi-Moussa and Yoccoz in genus two. The talk is based on joint works with Selim Ghazouani.