Ações Anosov de contato uniformemente quaseconformes

Abstract

In this work we obtain rigidity results for Rk Anosov actions in order to obtain positive answers in the direction of the Katok-Spatzier Conjecture: Every irreducible smooth Anosov action of R k or Z k (k g 2) is C ∞-conjugated to an algebraic action. More precisely, our main result (Theorem A) states that “all uniformly u-quasiconformal Anosov action associated with a generalized k-contact struture in a closed (2n + k)-manifold (n g 2) is C ∞-conjugated to a quasi-algebraic Anosov action”. This generalizes a result obtained by Sadovskaya [46] for Anosov flows. To prove this theorem, two fundamental results stand out, in the first one (Theorem B) we prove that an irreducible and uniformly uquasiconformal Anosov action is u-conformal with respect to a Riemannian metric (H¨older continuous and smooth on strong leaves) in the strong unstable distribution. Finally, in the second (Theorem C) we prove that uniformly u-quasiconformal Anosov actions associated with a generalized k-contact struture have smooth strong unstable foliation and strong stable foliation.