Boris Kalinin

Rigidity of the measurable structure for algebraic actions of higher-rank Abelian groups

We investigate rigidity of measurable structure for higher-rank Abelian algebraic actions. In particular, we show that ergodic measures for these actions fiber over a zero entropy measure with Haar measures along the leaves. We deduce various rigidity theorems for isomorphisms and joinings as corollaries.

GLOBAL RIGIDITY OF HIGHER RANK ANOSOV ACTIONS ON TORI AND NILMANIFOLDS

We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.

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.

Measurable rigidity and disjointness for Z k actions by toral automorphisms

We investigate joinings of strongly irreducible totally non-symplectic Anosov Z k , k ≥ 2 actions by toral automorphisms. We show that the existence of a non-trivial joining has strong implications for these actions, in particular, that the restrictions of the actions to a finite index subgroup of Z k are conjugate over Q. We also obtain a description of the joining measures modulo the classification of zero entropy measures for the actions.

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.

Totally nonsymplectic Anosov actions on tori and nilmanifolds

We show that sufficiently irreducible totally non-symplectic Anosov actions of higher rank abelian groups on tori and nilmanifolds are C 1 -conjugate to actions by affine automorphisms.

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.

On pointwise dimension of non-hyperbolic measures

We construct a diffeomorphism preserving a non-hyperbolic measure whose pointwise dimension does not exist almost everywhere. In the one-dimensional case we also show that such diffeomorphisms are typical in certain situations.

Lyapunov exponents of cocycles over non-uniformly hyperbolic systems

We consider linear cocycles over non-uniformly hyperbolic dynamical systems. The base system is a diffeomorphism \begin{document}

Livšic Theorem for matrix cocycles

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