We prove the Livšic Theorem for arbitrary GL(m, R) cocycles. We consider a hyperbolic dynamical system f : X → X and a Hölder continuous function A : X → GL(m, R). We show that if A has trivial periodic data, i.e. A(f n−1 p) · · · A(f p)A(p) = Id for each periodic point p = f n p, then there exists a Hölder continuous function C : X → GL(m, R) satisfying A(x) = C(f x)C(x) −1 for all x ∈ X. The main new ingredients in the proof are results of independent interest on relations between the periodic data, Lyapunov exponents, and uniform estimates on growth of products along orbits for an arbitrary Hölder function A.
Abstract. The first part of the paper begins with an introduction into Anosov actions of Z k and R k and an overview of the method of studying invariant measures for such actions based on consideration of conditional measures along various invariant foliations. The main body of that part contains a detailed proof of a modified version of the main theorem from [KS3] for actions by toral automorphisms of with applications to rigidity of the measurable structure of such actions with respect to Lebesque measure. In the second part principal technical tools for studying nonuniformly hyperbolic actions of Z k and R k are introduced and developed. These include Lyapunov characteristic exponents, nonstationary normal forms and Lyapunov Hoelder structures. At the end new rigidity results for Z 2 actions on three-dimensional manifolds are outlined.In this paper we discuss various results concerning invariant measures for actions of higher-rank abelian groups, i.e. Z k and R k for k ≥ 2 on compact differentiable manifolds which display certain hyperbolic behavior. Similarly to the rank one case, hyperbolicity can be full or partial, and uniform or nonuniform. Full (corr. partial) uniform hyperbolicity appears for Anosov (corr. partially hyperbolic) actions. Nonuniform hyperbolicity appears for actions preserving measures for which all (for the full case) or some (for partial case) Lyapunov characteristic exponents do not vanish. Two parts of the paper deals with the uniform and nonuniform cases correspondingly.While the final text of this paper is the product of a joint effort the basic drafts of various parts were written separartely: that for Section 3 was written by B. Kalinin and based on a part of his Ph. D thesis; for the rest of the paper the draft was written by A. Katok.
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.
We study higher-rank Cartan actions on compact manifolds preserving an ergodic measure with full support. In particular, we classify actions by R k with k ≥ 3 whose one-parameter groups act transitively as well as nondegenerate totally nonsymplectic Z k -actions for k ≥ 3.
Abstract. We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are C ∞ -conjugate to affine actions.
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