Abstract. We consider Hölder continuous linear cocycles over partially hyperbolic diffeomorphisms. For fiber bunched cocycles with one Lyapunov exponent we show continuity of measurable invariant conformal structures and sub-bundles. Further, we establish a continuous version of Zimmer's Amenable Reduction Theorem. For cocycles over hyperbolic systems we also obtain polynomial growth estimates for the norm and quasiconformal distortion from the periodic data.

Abstract. We show that for any uniformly quasiconformal symplectic Anosov diffeomorphism of a compact manifold of dimension at least 4, its finite cover is C ∞ conjugate to an Anosov automorphism of a torus. We also prove that any uniformly quasiconformal contact Anosov flow on a compact manifold of dimension at least 5 is essentially C ∞ conjugate to the geodesic flow of a manifold of constant negative curvature.

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.

We consider an irreducible Anosov automorphism L of a torus T d such that no three eigenvalues have the same modulus. We show that L is locally rigid, that is, L is C 1+Hölder conjugate to any C 1 -small perturbation f such that the derivative D p f n is conjugate to L n whenever f n p = p. We also prove that toral automorphisms satisfying these assumptions are generic in SL(d, Z). Examples constructed in the Appendix show importance of the assumption on the eigenvalues. F. Voloch. Unit in a number field with same at a real and a complex place. URL:

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