A Note on Rigidity of Anosov diffeomorphisms of the Three Torus

2018/05/31 by Micena, F., Tahzibi, A.
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1805.12288

Abstract

We consider Anosov diffeomorphisms on \mathbbT³ such that the tangent bundle splits into three subbundles E^s_f ⊕ E^wu_f ⊕ E^su_f. We show that if f is C^r, r ≥ 2, volume preserving, then f is C¹ conjugated with its linear part A if and only if the center foliation F^wu_f is absolutely continuous and the equality λ^wu_f(x) = λ^wu_A, between center Lyapunov exponents of f and A, holds for m a.e. x ∈ \mathbbT³. 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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