Anosov diffeomorphisms, renormalization and tilings

Abstract

In this thesis, we prove a one-to-one correspondence between C^{1+} smooth conjugacy classes of circle diffeomorphisms that are C^{1+} fixed points of renormalization and C^{1+} conjugacy classes of Anosov diffeomorphisms whose Sinai-Ruelle-Bowen measure is absolutely continuous with respect to Lebesgue measure. Furthermore, we use ratio functions to parametrize the infinite dimensional space of C^{1+} smooth conjugacy classes of circle diffeomorphisms that are C^{1+} fixed points of renormalization. We introduce the notion of γ-tilings and we prove a one-to-one correspondence between (i) smooth conjugacy classes of Anosov diffeomorphisms, with an invariant measure absolutely continuous with respect to the Lebesgue measure, (ii) affine classes of γ-tilings and (iii) solenoid functions. The solenoid functions give a parametrization of the infinite dimensional space consisting of the mathematical objects described in the above equivalences.