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Positive entropy actions by higher-rank lattices

Aaron Brown, Homin Lee

TL;DR

The paper proves rigidity for smooth actions of higher-rank lattices under the mild hypothesis that some element has positive topological entropy. It builds a suspension space framework to translate topological entropy into fiber-entropy properties, uses higher-rank structure to force extra invariance, and leverages cocycle superrigidity to obtain measurable conjugacies to affine actions on (infra-)tori. The main results show commensurability with $SL(n,\mathbb{Z})$ for positive-entropy actions, existence of AC invariant measures with positive entropy, and, in favorable cases, measurable conjugacy to affine toral actions; for Anosov-type actions, the manifold is forced to be a torus and the action affine. Collectively, the results advance Zimmer-type rigidity by connecting entropy, suspension dynamics, and homogeneous-measure classifications to deduce strong structural conclusions about the action and the manifold.

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.

Positive entropy actions by higher-rank lattices

TL;DR

The paper proves rigidity for smooth actions of higher-rank lattices under the mild hypothesis that some element has positive topological entropy. It builds a suspension space framework to translate topological entropy into fiber-entropy properties, uses higher-rank structure to force extra invariance, and leverages cocycle superrigidity to obtain measurable conjugacies to affine actions on (infra-)tori. The main results show commensurability with for positive-entropy actions, existence of AC invariant measures with positive entropy, and, in favorable cases, measurable conjugacy to affine toral actions; for Anosov-type actions, the manifold is forced to be a torus and the action affine. Collectively, the results advance Zimmer-type rigidity by connecting entropy, suspension dynamics, and homogeneous-measure classifications to deduce strong structural conclusions about the action and the manifold.

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 acting on -manifolds, if the action has positive topological entropy we show the lattice must be commensurable with . 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.
Paper Structure (90 sections, 70 theorems, 214 equations, 3 tables)

This paper contains 90 sections, 70 theorems, 214 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Main results for actions by lattices in $\mathrm{SL}(n,\mathbb {R})$
  3. Motivation: affine $\mathrm{SL}(n,\mathbb {Z})$-actions and their deformations
  4. Classification of lattices acting with positive entropy
  5. Actions with positive entropy admit absolutely continuous invariant measures
  6. Measurable classification of actions with positive entropy; proof of \ref{['coro:SLn']}
  7. Consequences for Anosov actions
  8. Topological consequences
  9. Actions by lattices in split orthogonal groups
  10. Suspension space, two themes, and formulation of main technical results
  11. Standing hypotheses on $G$ and $\Gamma$ and reductions
  12. Suspension, induced $G$-action, and fiber entropy
  13. Suspension space and induced $G$-action
  14. Fiber entropy and conditional measures
  15. Dictionary between invariant measures
  16. ...and 75 more sections

Key Result

Corollary 1

[corollary]coro:SLn For $n\ge 3$, let $\Gamma$ be a lattice in $\mathrm{SL}(n,{\mathbb R})$. Let $M$ be a closed manifold with $\dim M= n$ and let $\alpha\colon\Gamma\to\mathrm{Diff}^\infty(M)$ be an action such that $h_{\mathrm{top}}(\alpha(\gamma_{0}))>0$ for some $\gamma_{0}\in \Gamma$. Then $\Ga

Theorems & Definitions (117)

  • Corollary 1
  • Corollary 2
  • Theorem 1.1
  • Theorem : HL:Dom
  • Corollary 3
  • Remark 1
  • Remark 2: Anosov actions of $\Gamma<\mathrm{SL}(n,\mathbb {R})$ for $n=3$
  • Corollary 4
  • Corollary 5
  • proof : Proof of \ref{['coro:SO']}
  • ...and 107 more