Entropy, Lyapunov exponents, and rigidity of group actions

Abstract

This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled Workshop for young researchers: Groups acting on manifolds held in Teresopolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory-particularly Lya-punov exponents and metric entropy-as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelian actions on tori due to A. Katok and R. Spatzier [114] and recent the work of the author with D. Fisher, S. Hurtado, F. Rodriguez Hertz, and Z. Wang on actions of lattices in higher-rank semisimple Lie groups on manifolds [31, 36]. We give complete proofs of these results and present sufficient background in smooth ergodic theory needed for the proofs. A unifying theme in this text is the use of metric entropy and its relation to the geometry of conditional measures along foliations as a mechanism to verify invariance of measures. i