Victoria Sadovskaya

Global rigidity for totally nonsymplectic Anosov Z k actions

We consider a totally nonsymplectic Anosov action of Z^k which is either uniformly quasiconformal or pinched on each coarse Lyapunov distribution. We show that such an action on a torus is C^\infty--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 jointly integrable.

Cohomology of fiber bunched cocycles over hyperbolic systems

We consider Holder continuous fiber bunched GL(d,R)-valued cocycles over an Anosov diffeomorphism. We show that two such cocycles are Holder continuously cohomologous if they have equal periodic data, and prove a result for cocycles with conjugate periodic data. We obtain a corollary for cohomology between any constant cocycle and its small perturbation. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base. We show that this condition can be established based on the periodic data. Some important examples of cocycles come from the differential of the diffeomorphism and its restrictions to invariant sub-bundles. We discuss an application of our results to the question when an Anosov diffeomorphism is smoothly conjugate to a C^1-small perturbation. We also establish Holder continuity of a measurable conjugacy between a fiber bunched cocycle and a uniformly quasiconformal one. Our main results also hold for cocycles with values in a closed subgroup of GL(d,R), for cocycles over hyperbolic sets and shifts of finite type, and for linear cocycles on a non-trivial vector bundle.

ON LOCAL AND GLOBAL RIGIDITY OF QUASI-CONFORMAL ANOSOV DIFFEOMORPHISMS

We consider a transitive uniformly quasi-conformal Anosov diffeomorphism f of a compact manifold M. We prove that if the stable and unstable distributions have dimensions greater than two, then f is C ∞ conjugate to an affine Anosov automorphism of a finite factor of a torus. If the dimensions are at least two, the same conclusion holds under the additional assumption that M is an infranilman- ifold. We also describe necessary and sufficient conditions for smoothness of conjugacy between such a diffeomorphism and a small perturbation.

On Anosov diffeomorphisms with asymptotically conformal periodic data

We consider transitive Anosov diffeomorphisms for which every periodic orbit has only one positive and one negative Lyapunov exponent. We prove various properties of such systems, including strong pinching, C 1C smoothness of the Anosov splitting, and C 1 smoothness of measurable invariant conformal structures and distributions. We apply these results to volume-preserving diffeomorphisms with two-dimensional stable and unstable distributions and diagonalizable derivatives of the return maps at periodic points. We show that a finite cover of such a diffeomorphism is smoothly conjugate to an Anosov automorphism of T 4 ; as a corollary, we obtain local rigidity for such diffeomorphisms. We also establish a local rigidity result for Anosov diffeomorphisms in dimension three.

Dimensional characteristics of invariant measures for circle diffeomorphisms

We consider pointwise, box, and Hausdorff dimensions of invariant measures for circle diffeomorphisms. We discuss the cases of rational, Diophantine, and Liouville rotation numbers. Our main result is that for any Liouville number

On pointwise dimension of non-hyperbolic measures

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Lyapunov exponents of cocycles over non-uniformly hyperbolic systems

there exists a

Multifractal Analysis of Conformal Axiom A Flows

C^\infty

On Uniformly Quasiconformal Anosov Systems

circle diffeomorphism with rotation number

LINEAR COCYCLES OVER HYPERBOLIC SYSTEMS AND CRITERIA OF CONFORMALITY

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