This paper is a survey about recent progress in measure rigidity and global rigidity of Anosov actions, and celebrates the profound contributions by Federico Rodriguez Hertz to rigidity in dynamical systems.
In @cite_29 , A. Katok and F. Rodriguez Hertz obtained the classification of a real analytic action @math of @math on the @math -torus @math assuming the induced map on homology is standard: on the complement of a finite set, there is a real analytic conjugacy from the linearized action to @math whose image has full measure. Very recently, A. Brown, F. Rodriguez Hertz and Z. Wang found a major improvement @cite_9 . We will discuss it below in Section .
@cite_9 @cite_29
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In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose @math is a lattice in semisimple Lie group, all of whose factors have rank @math or higher. Let @math be a smooth @math -action on a compact nilmanifold @math that lifts to an action on the universal cover. If the linear data @math of @math contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of @math and @math , on a finite-index subgroup of @math . If @math is a @math action and contains an Anosov element, then the semiconjugacy is a @math conjugacy. As a corollary, we obtain @math global rigidity for Anosov actions by cocompact lattices in semisimple Lie group with all factors rank @math or higher. We also obtain global rigidity of Anosov actions of @math on @math for @math and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds. We prove that any real-analytic action of @math with standard homotopy data that preserves an ergodic measure @math whose support is not contained in a ball, is analytically conjugate on an open invariant set to the standard linear action on the complement to a finite union of periodic orbits.