Arithmeticity for Smooth Maximal Rank Positive Entropy Actions of $\mathbb{R}^k$

Abstract

We establish arithmeticity in the sense of A. Katok and F. Rodriguez Hertz of smooth actions $α$ of $\mathbb{R}^k$ 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 $\mathbb{Z}^k$. This in particular solves, up to measure theoretical isomorphism, Problem 4 from a prequel paper of Katok and Rodriguez Hertz, joint with B. Kalinin.