A Note on Rigidity of Anosov diffeomorphisms of the Three Torus

Abstract

We consider Anosov diffeomorphisms on ${\mathbb{T}}^3$ such that the tangent bundle splits into three subbundles $E_f^s\oplus E_f^{wu}\oplus E_f^{su}.$ We show that if $f$ is $C^r,r\ge 2,$ volume preserving, then $f$ is $C^1$ conjugated with its linear part $A$ if and only if the center foliation ${\mathcal{F}}_f^{wu}$ is absolutely continuous and the equality ${\lambda }_f^{wu}(x)={\lambda }_A^{wu},$ 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.