Local rigidity of toral automorphisms

I will discuss rank one rigidity for hyperbolic toral automorphisms. Classically such rigidity results rely on coincidence of eigendata at periodic points. However, one can of course consider other invariant measures. Recently Saghin and Yang established local rigidity from coincidence of volume Lyapunov exponents. One remarkable consequence of this breakthrough is that one can now address partially hyperbolic setting (where periodic points do not exist for granted). This approach was further pursued by myself, Kalinin and Sadovskaya.