Global rigidity for totally nonsymplectic Anosov ℤ^kactions

Abstract

We consider a totally nonsymplectic (TNS) Anosov action of ‫ޚ‬ k which is either uniformly quasiconformal or pinched on each coarse Lyapunov distribution. We show that such an action on a torus is C 1 -conjugate to an action by affine automorphisms. We also obtain similar global rigidity results for actions on an arbitrary compact manifold assuming that the coarse Lyapunov foliations are topologically jointly integrable. 37C15, 37D99

Global rigidity of higher rank Anosov actions on tori and nilmanifolds

“…By commutativity, the Lyapunov decompositions for individual elements of Z k can be refined to a joint invariant splitting for the action. The following proposition from [22] describes the Multiplicative Ergodic Theorem for this case. See [20] for more details on the Multiplicative Ergodic Theorem and related notions for higher rank abelian actions.…”

Diffeomorphism group valued cocycles over higher-rank abelian Anosov actions

“…In the presence of sufficiently many Anosov elements (an Anosov element in each Weyl chamber) and if the invariant measure is of full support (such a measure always exists if there is a transitive Anosov element in the action) the coarse Lyapunov distributions are intersections of stable distributions for various elements of the action, they are well defined everywhere, Hölder continuous, and tangent to foliations with smooth leaves. (For more details see Section 2 in [20] or Section 2.2 in [21]). Moreover, for any other action invariant measure of full support, and Anosov elements in each Weyl chamber, the coarse Lyapunov distributions will be the same, as well as the Weyl chamber picture.…”

On the work of Rodriguez Hertz on rigidity in dynamics

“…by A. Katok and J. Lewis [KL91,Theorem 4.12], under stringent assumptions such as onedimensionality of the coarse Lyapunov foliations, or the classification of Cartan and conformal Anosov actions of R k and Z k on general manifolds, cf. B. Kalinin and R. Spatzier [KS07b] for k ≥ 3 and B. Kalinin and V. Sadovskaya [KS06,KS07a]. Kalinin and Sadovskaya recently announced a generalization of the classification for Cartan actions to Z 2 and R 2 .…”