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arXiv · 1805.12288

A Note on Rigidity of Anosov diffeomorphisms of the Three Torus

F. Micena·A. Tahzibi
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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.

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F. Micena, A. Tahzibi. 2018-05-31. A Note on Rigidity of Anosov diffeomorphisms of the Three Torus. https://arxiv.org/abs/1805.12288

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