By Alp Uzman
Appears in collection : 2024 - T2 - WS3 - Actions of large groups, geometric structures, and the Zimmer program
We prove an arithmeticity theorem in the context of nonuniform measure rigidity. Adapting machinery developed by A. Katok and F. Rodriguez Hertz [J. Mod. Dyn. 10 (2016), 135–172; MR3503686] for $\mathbb{Z}^k$ systems to $\mathbb{R}^k$ systems, we show that any maximal rank positive entropy system on a manifold generated by $k>=2$ commuting vector fields of regularity $\mathbb{C}^k$ for $r>1$ is measure theoretically isomorphic to a constant time change of the suspension of some action of $\mathbb{Z}^k$ on the $(k+1)$-torus or the $(k+1)$-torus modulo {id,-id} by affine automorphisms with linear parts hyperbolic. Further, the constructed conjugacy has certain smoothness properties. This in particular answers a problem and a conjecture from a prequel paper of Katok and Rodriguez Hertz, joint with B. Kalinin [Ann. of Math. (2) 174 (2011), no. 1, 361–400; MR2811602].
Alp Uzman : Arithmeticity for Smooth Maximal Rank Positive Entropy Actions of $\mathbb{R}^k$, 2023. arxiv:2307.09433
By Anna Florio
By Sven Sandfeldt
By Cyril Houdayer
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