Journal of Modern Dynamics (JMD)

Exponential mixing and smooth classification of commuting expanding maps

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

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Ralf Spatzier - Department of Mathematics, 2074 East Hall, 530 Church Street, University of Michigan, Ann Arbor, MI 48109-1043, United States (email)
Lei Yang - Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 9190401, Israel (email)

Abstract: 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.

Keywords:  Higher rank expanding actions, smooth classification, nilmanifolds, exponential mixing.
Mathematics Subject Classification:  Primary: 37C15, 37C85, 37D20, 53C24; Secondary: 42B05.

Received: August 2016;      Revised: February 2017;      Available Online: March 2017.