Global smooth and topological rigidity of hyperbolic lattice actions

In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices.
Suppose $\Gamma$ is a lattice in a semisimple Lie group, all of whose factors have rank $2$ or higher. Let $\alpha$ be a smooth $\Gamma$-action on a compact nilmanifold $M$ that lifts to an action on the universal cover. If the linear data $\rho$ of $\alpha$ contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of $\alpha$ and $\rho$ on a finite-index subgroup of $\Gamma$. If $\alpha$ is a $C^\infty$ action and contains an Anosov element, then the semiconjugacy is a $C^\infty$ conjugacy.
As a corollary, we obtain $C^\infty$ global rigidity for Anosov actions by cocompact lattices in semisimple Lie groups with all factors rank $2$ or higher. We also obtain global rigidity of Anosov actions of $\mathrm{SL}(n,\mathbb{Z})$ on $\mathbb{T}^n$ for $ n\geq 5$ and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds.

Pages 913-972 by Aaron Brown, Federico Rodriguez Hertz, Zhiren Wang | From volume 186-3

Nonuniform measure rigidity

We consider an ergodic invariant measure $\mu$ for a smooth action $\alpha$ of $\mathbb{Z}^k$, $k\ge 2$, on a $(k+1)$-dimensional manifold or for a locally free smooth action of $\mathbb{R}^k$, $k\ge 2$, on a $(2k+1)$-dimensional manifold. We prove that if $\mu$ is hyperbolic with the Lyapunov hyperplanes in general position and if one element in $\mathbb{Z}^k$ has positive entropy, then $\mu$ is absolutely continuous. The main ingredient is absolute continuity of conditional measures on Lyapunov foliations which holds for a more general class of smooth actions of higher rank abelian groups.

Pages 361-400 by Boris Kalinin, Anatole Katok, Federico Rodriguez Hertz | From volume 174-1

Stable ergodicity of certain linear automorphisms of the torus

We find a class of ergodic linear automorphisms of $\mathbb{T}^N$ that are stably ergodic. This class includes all non-Anosov ergodic automorphisms when $N=4$. As a corollary, we obtain the fact that all ergodic linear automorphism of $\mathbb{T}^N$ are stably ergodic when $N\leq 5$.

Pages 65-107 by Federico Rodriguez Hertz | From volume 162-1