A Note on Rigidity of Anosov diffeomorphisms of the Three Torus

TL;DR

This paper investigates conditions under which volume-preserving Anosov diffeomorphisms on the three-torus are smoothly conjugate to their linear models, focusing on foliation regularity and Lyapunov exponents.

Contribution

It establishes a characterization of smooth conjugacy for three-dimensional Anosov diffeomorphisms based on foliation absolute continuity and Lyapunov exponent equality.

Findings

Conjugacy to linear models occurs when the center foliation is absolutely continuous.

Equality of center Lyapunov exponents with the linear model characterizes conjugacy.

Strong absolute continuity of stable and unstable foliations implies rigidity of derived from Anosov diffeomorphisms.

Abstract

We consider Anosov diffeomorphisms on $\mathbb{T}^3$ such that the tangent bundle splits into three subbundles $E^s_f \oplus E^{wu}_f \oplus E^{su}_f.$ We show that if $f$ is $C^r, r \geq 2,$ volume preserving, then $f$ is $C^1$ conjugated with its linear part $A$ if and only if the center foliation $\mathcal{F}^{wu}_f$ is absolutely continuous and the equality $\lambda^{wu}_f(x) = \lambda^{wu}_A,$ between center Lyapunov exponents of $f$ and $A,$ holds for $m$ a.e. $x \in \mathbb{T}^3.$ We also conclude rigidity of derived from Anosov diffeomorphism, assuming an strong absolute continuity property (Uniform bounded density property) of strong stable and strong unstable foliations.