Smooth Stable Foliations of Anosov Diffeomorphisms

TL;DR

This paper proves that for Anosov diffeomorphisms, stable foliations with slightly more than twice differentiability are actually as smooth as the diffeomorphisms themselves, highlighting a rigidity property.

Contribution

It establishes a new rigidity result for $C^{2+}$-smooth stable foliations of Anosov diffeomorphisms, showing they match the smoothness of the diffeomorphisms.

Findings

Stable foliations with regularity > 2 are as smooth as the diffeomorphisms

Rigidity of smoothness levels for Anosov systems

Enhancement of regularity results in hyperbolic dynamics

Abstract

In this paper, we focus on the rigidity of $C^{2+}$-smooth codimension-one stable foliations of Anosov diffeomorphisms. Specifically, we show that if the regularity of these foliations is slightly bigger than $2$, then they will have the same smoothness as the diffeomorphisms.