Entropy, Smooth Ergodic Theory, and Rigidity of Group Actions

This is a preliminary version of lecture notes based on a 6 hour course given at the workshop Dynamics Beyond Uniform Hyperbolicity held in Provo, Utah June 2017. The goal of this course is two-fold. First we will present a number of tools and results in the smooth ergodic theory of actions of higher-rank abelian groups including Lyapunov exponents, metric entropy, and the relationship between entropy, exponents, and geometry of conditionalmeasures. We will explain the main proposition (the invariance principle that “non-resonance implies invariance”) from the work of Brown-Rodriguez Hertz-Wang and explain their main theorem: every action of a lattice in SL(n,R) on an (n−2)-dimensional manifold preserves a Borel probability measure. Second, we will outline the proof of the main theorem of Brown-Fisher-Hurtado: every action of a cocompact lattice in SL(n,R) on a (n− 2)-dimensional manifold is finite. We will explain some of the main tools used in the proof: Strong property (T), Margulis superrigidity, and Zimmer’s cocycle superrigidity and how they combine with the invariance principle from the work of Brown-Rodriguez Hertz-Wang and Ratner’s theorems to prove the theorem.