Positive entropy actions by higher-rank lattices

TL;DR

This paper proves new rigidity results for smooth actions of higher-rank lattice groups on manifolds, showing positive entropy actions are closely related to algebraic and affine models, with implications for measure and entropy properties.

Contribution

It establishes that positive entropy actions by higher-rank lattices are essentially algebraic or affine, extending rigidity results and analyzing entropy behavior under limits.

Findings

Lattices in SL(n,R) with positive entropy actions are commensurable with SL(n,Z)

Such actions admit absolutely continuous measures with positive metric entropy

Actions are measurably conjugate to affine actions on tori

Abstract

We study smooth actions by lattices in higher-rank simple Lie groups. Assuming one element of the action acts with positive topological entropy, we prove a number of new rigidity results. For lattices in $\mathrm{SL}(n,\mathbb{R})$ acting on $n$-manifolds, if the action has positive topological entropy we show the lattice must be commensurable with $\mathrm{SL}(n,\mathbb{Z})$. Moreover, such actions admit an absolutely continuous probability measure with positive metric entropy; adapting arguments by Katok and Rodriguez Hertz, we show such actions are measurably conjugate to affine actions on (infra-)tori. In a main technical argument, we study families of probability measures invariant under sub-actions of the induced action by the ambient Lie group on an associated fiber bundle. To control entropy properties of such measures when passing to limits, in the appendix we establish certain upper semicontinuity of fiber entropy under weak-$*$ convergence, adapting classical results of Yomdin and Newhouse.