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

TL;DR

This paper proves that certain smooth actions of rb1^k with positive entropy are measure-theoretically equivalent to affine actions, establishing a form of arithmeticity for these dynamical systems.

Contribution

It demonstrates that smooth rb1^k actions with positive entropy are measure-theoretically isomorphic to affine Cartan actions, solving a previously open problem.

Findings

Shows measure-theoretic isomorphism to affine actions

Establishes arithmeticity for smooth rb1^k actions

Addresses a key open problem in the field

Abstract

We establish arithmeticity in the sense of A. Katok and F. Rodriguez Hertz of smooth actions $\alpha$ 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 $\alpha_t$ of the action has positive entropy. Arithmeticity in this context means that the action $\alpha$ 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.