There exist no contact Anosov diffeomorphisms

TL;DR

This paper proves that on closed odd-dimensional manifolds, no Anosov diffeomorphism can preserve an invariant contact structure, highlighting a fundamental incompatibility between these dynamical systems and contact geometry.

Contribution

The paper establishes a non-existence result for invariant contact structures under Anosov diffeomorphisms on odd-dimensional manifolds.

Findings

No invariant contact structures exist for Anosov diffeomorphisms on closed odd-dimensional manifolds.

The result constrains the types of dynamical systems compatible with contact geometry.

Provides a new understanding of the interplay between hyperbolic dynamics and geometric structures.

Abstract

For any Anosov diffeomorphims on a closed odd dimensional manifold, there exists no invariant contact structure.