2024/09/09 by Aaron Brown, Brown, Aaron, Homin Lee +1
Computer Science · #Advanced Algebra and Logic #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2409.05991
openalex publication_date 2024/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 SL(n,ℝ) acting on n-manifolds, if the action has positive topological entropy we show the lattice must be commensurable with SL(n,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.