☆ 4.2 Article

EXPONENTIAL MIXING AND SMOOTH CLASSIFICATION OF COMMUTING EXPANDING MAPS

JOURNAL OF MODERN DYNAMICS (2017)

Journal

JOURNAL OF MODERN DYNAMICS

Volume 11, Issue -, Pages 263-312

Publisher

AMER INST MATHEMATICAL SCIENCES-AIMS

DOI: 10.3934/jmd.2017012

Keywords

Higher rank expanding actions; smooth classification; nilmanifolds exponential mixing

Categories

Mathematics, Applied Mathematics

Funding

NSF [DMS-1307164]
MSRI
ERC [AdG 267259]
ISF [0399180]
Direct For Mathematical & Physical Scien
Division Of Mathematical Sciences [1307164] Funding Source: National Science Foundation

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Abstract

We show that genuinely higher rank expanding actions of abelian semigroups on compact manifolds are C-infinity -conjugate to affine actions on infra-nilmanifolds. This is based on the classification of expanding diffeomorphisms up to Holder 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.

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