Exponential Mixing and Smooth Classification of Commuting Expanding Maps

TL;DR

This paper proves that higher rank expanding actions of abelian semi-groups on compact manifolds are smoothly conjugate to affine actions on infra-nilmanifolds, using exponential mixing and classification results.

Contribution

It establishes smooth conjugacy for higher rank expanding actions, extending classification techniques and proving exponential mixing of solenoid actions.

Findings

Higher rank expanding actions are $C^{ abla}$-conjugate to affine actions.

Exponential mixing of solenoid actions is established.

The regularity of conjugacy is proven using mixing results.

Abstract

We show that genuinely higher rank expanding actions of abelian semi-groups on compact manifolds are $C^{\infty}$-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 semi-group actions by nilmanifold endomorphisms, a result of independent interest. We then proceed similar to the case of higher rank Anosov actions.