Entropy, Lyapunov exponents, and rigidity of group actions

TL;DR

This paper explores the use of entropy and Lyapunov exponents in understanding the rigidity of higher-rank group actions on manifolds, providing detailed proofs of key theorems in smooth ergodic theory.

Contribution

It offers complete proofs of measure rigidity results for affine Anosov actions and lattice actions, with a focus on entropy and geometric measure invariance.

Findings

Proof of measure rigidity for affine Anosov abelian actions

Rigidity results for lattice actions in higher-rank semisimple Lie groups

Use of metric entropy to verify measure invariance along foliations

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 Teres\'opolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory -- particularly Lyapunov 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 [Ergodic Theory Dynam. Systems 16, 1996] and recent the work of the main author with D. Fisher, S. Hurtado, F. Rodriguez Hertz, and Z. Wang on actions of lattices in higher-rank semisimple Lie groups on manifolds [arXiv:1608.04995; arXiv:1610.09997]. 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.