Arithmeticity for Smooth Maximal Rank Positive Entropy Actions of ℝ^k

2023/07/18 by Alp Uzman, Uzman, Alp
Mathematics · #37A35 #37C40 #37C85 #37D25 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2307.09433

openalex publication_date 2023/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish arithmeticity in the sense of A. Katok and F. Rodriguez Hertz of smooth actions α of ℝ^k on an anonymous manifold M of dimension 2k+1 provided that there is an ergodic invariant Borel probability measure on M w/r/t which each nontrivial time-t map α_t of the action has positive entropy. Arithmeticity in this context means that the action α is measure theoretically isomorphic to a constant time change of the suspension of an affine Cartan action of ℤ^k. This in particular solves, up to measure theoretical isomorphism, Problem 4 from a prequel paper of Katok and Rodriguez Hertz, joint with B. Kalinin.

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