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

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.