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Title:
Convergence (and divergence) of renormalization in higher genus
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Abstract:
As circle diffeomorphisms (and circle diffeomorphisms with breakpoints) 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). Renormalization on the space of GIETs presents many similarities with the classical theory of circle diffeos and circle diffeos with break points, but also many crucial differences and new challenges (notably, the lack of a Denjoy-Koksma inequality and 'a priori' bounds). In the talk we will first try to briefly highlight both similarities and differences and then present a recent result, in which we prove that, under suitable arithmetic conditions, the orbits under renormalization satisfies a dynamical dichotomy which distinguish between convergent and divergent behavior. In genus two, we are then able to exploit this result to 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 conjugate to it. This is joint work with Selim Ghazouani.
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