# American Institute of Mathematical Sciences

2017, 11: 263-312. doi: 10.3934/jmd.2017012

## Exponential mixing and smooth classification of commuting expanding maps

 1 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA 2 Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 9190401, Israel

Received  August 07, 2016 Revised  February 04, 2017 Published  March 2017

Fund Project: RS: Supported in part by NSF grant DMS-1307164 and an Eisenbud Professorship at MSRI.
LY: Supported in part by a Postdoctoral Fellowship at MSRI, ERC grant AdG 267259, and ISF grant 0399180.

We show that genuinely higher rank expanding actions of abelian semigroups on compact manifolds are $C.{\infty}$-conjugate to affine actions on infra-nilmanifolds. This is based on the classification of expanding diffeomorphisms up to Hölder conjugacy by Gromov and Shub, and is similar to recent work on smooth classification of higher rank Anosov actions on tori and nilmanifolds. To prove regularity of the conjugacy in the higher rank setting, we establish exponential mixing of solenoid actions induced from semigroup actions by nilmanifold endomorphisms, a result of independent interest. We then proceed similar to the case of higher rank Anosov actions.

Citation: Ralf Spatzier, Lei Yang. Exponential mixing and smooth classification of commuting expanding maps. Journal of Modern Dynamics, 2017, 11: 263-312. doi: 10.3934/jmd.2017012
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