The aim of this class is to show some basic ideas of D. Fisher, B. Kalinin and R. Spatzier 's proof for Katok-Spatzier conjecture on torus, Global Rigidity of Higher Rank Anosov Actions on Tori and Nilmanifolds, Journal of the American Mathematical Society v. 26 (2013), 167-198. The key part of their proof is elegant and contains some very interesting feature.

This class will begin with the definition of some basic properties of Anosov diffeomorphisms, e.g. Franks-Manning conjugacy and then introduce Lyapunov exponents, classical Oseledec theorem and their Z^k version, we will also mention the Weyl chamber picture and its relationship with representation theory. Later we move on to the Katznelson's lemma and exponential mixing property for commuting toral automorphisms. From that point on, this class will introduce a useful result for regularity of functions along absolutely continuous foliation, which is a variation of results of Rauch and Taylor, and then we could go through the key part of the proof.


Some basic knowledge on differential geometry, Fourier analysis on tori and ergodic theory.