On a Smale Conjecture for the existence of fixed points for Anosov diffeomorphisms

TL;DR

This paper proves that Anosov diffeomorphisms with sufficiently smooth stable and unstable foliations on compact manifolds must have fixed points, partially confirming a conjecture by Smale.

Contribution

It establishes a new condition involving foliation smoothness that guarantees fixed points for Anosov diffeomorphisms, advancing understanding of their fixed point properties.

Findings

Anosov diffeomorphisms with $C^3$ foliations have fixed points

Provides a partial positive answer to Smale's conjecture

Highlights the role of foliation smoothness in fixed point existence

Abstract

We prove that if the stable foliation and the unstable foliation of an Anosov diffeomorphism on a connected compact manifold are $C^3$, then the diffeomorphism has fixed points. This is a partial positive answer to a Smale conjecture for fixed points of Anosov diffeomorphisms.