On classification of resonance-free Anosov ℤk actions

Abstract

We consider actions of Z k , k ≥ 2, by Anosov diffeomorphisms which are uniformly quasiconformal on each coarse Lyapunov distribution. These actions generalize Cartan actions for which coarse Lyapunov distributions are onedimensional. We show that, under certain non-resonance assumptions on the Lyapunov exponents, a finite cover of such an action is smoothly conjugate to an action by toral automorphisms.

Nonuniform measure rigidity

“…By contrast, global rigidity for Anosov actions on a torus [31] and nonuniform measure rigidity on a torus [7,20], as well as results of this paper, deal with maximal rank actions. Notice, however, that global rigidity results for Anosov actions on an arbitrary manifold satisfying stronger dynamical assumptions only require rank ≥ 3 [11] or rank ≥ 2 [9,10] We expect that global measure rigidity results, both on the torus and in the general setting similar to those of the present paper, hold at a greater generality although we do not see a realistic approach for the most general "genuine higher rank" situation even on the torus. There is still an intermediate class which is compatible with the lowest admissible rank (i.e rank two) on manifolds of arbitrary dimension.…”

Diffeomorphism group valued cocycles over higher-rank abelian Anosov actions

“…Since for any Weyl chamber, there is an element in PH(α), thereforẽ In the following proposition we prove the regulartiy ofẼ i along each coarse Lyapunov foliation. Our approach generalizes of the arguments in [19] to partially hyperbolic actions. Notice here that the quasiconformality assumptions in [19] are not used in our proof.…”

“…(cf. [19]) Since E is C r along W , then we can approximate E 1 and E 2 byĒ 1 and E 2 respectively such thatĒ 1,2 are subbundles of E and C r along W andĒ 1 ⊕Ē 2 is still a dominated splitting of E under df . Moreover we can assumẽ…”

“…For non-small cocycles the method we use for proving regularity of these foliations is inspired by our work in [6] on partially hyperbolic actions with compact center foliation. Rather than using leaf conjugacy from [10] we use the C r section theorem in a rather technical way, which can be viewed as an extension of argument in [19] to partially hyperbolic case.…”

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 .…”