Linearization and rigidity in higher genus

Speaker. Corinna Ulcigrai
Date. 31.01.23 at 2:00 pm

Abstract. Circle rotations and circle diffeomorphisms can be seen as Poincare maps of flows on tori. The maps that arise when considering Poincare maps of smooth flows on higher genus surfaces are piecewise continuous diffeomorphisms, known as generalized interval exchange maps (GIETs).

The study of linearization and rigidity questions (such as when a generalized IET is conjugated to its linear model or what is the regularity of the conjugacy) present similarities with the classical theory of circle diffeos, but also many crucial differences and new challenges.

In the talk we will give a brief survey of known results and open questions and then focus on past and recent joint work with Selim Ghazouani. In particular, in genus two, we prove geometric rigidity, by showing that, under a Diophantine-type condition, GIETs which is C^0 conjugate to their linear model are indeed C^1 (and C^{1+\apha}) conjugate to it.