06/07: Corinna Ulcigrai
Rigidity of foliations on surfaces and renormalization
Abstract: A class of dynamical systems is called (geometrically) rigid if the existence of a topological conjugacy implies automatically that the conjugacy is differentiable. Several classes of geometrically rigid system have been discovered in one-dimensional dynamics.
In particular, it follows from a celebrated result by Michel Herman on circle diffeomorphisms (later improved by Yoccoz) that minimal smooth orientable foliations on surfaces of genus one, under a full measure arithmetic condition, are geometrically rigid.
In very recent joint work with Selim Ghazouani, we prove a generalization of this result to genus two, in particular by showing that smooth, orientable foliations with non-degenerate (Morse) singularities on surfaces of genus two, under a full measure arithmetic condition, are geometrically rigid. This in particular proves the genus two case of a conjecture by Marmi, Moussa and Yoccoz (formulated in the language of the Poincare maps, namely generalized interval exchange transformations.
During the talk, after motivating and explaining the result, we will give a brief survey of some of the key results in the theory of circle diffeos and in the study of generalized interval exchange maps and then an brief overview the strategy of the proof, which is based on renormalization.