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Boris Kalinin

Publications and source records attributed to Boris Kalinin.

28 records · Page 2Linked to original sources
arXiv2010

Nonuniform measure rigidity

We consider an ergodic invariant measure $μ$ for a smooth action of $Z^k$, $k \ge 2$, on a $(k+1)$-dimensional manifold or for a locally free smooth action of $R^k$, $k \ge 2$ on a $(2k+1)$-dimensional manifold. We prove that if $μ$ is hyperbolic with the Lyapunov hyperplanes in general position and if one element of the action has positive entropy, then $μ$ is absolutely continuous. The main ingredient is absolute continuity of conditional measures on Lyapunov foliations which holds for a more general class of smooth actions of higher rank abelian groups.

math.DS↗
arXiv2010

Linear cocycles over hyperbolic systems and criteria of conformality

In this paper we study Holder continuous linear cocycles over transitive Anosov diffeomorphisms. Under various conditions of relative pinching we establish properties including existence and continuity of measurable invariant sub-bundles and conformal structures. We use these results to obtain criteria for cocycles to be isometric or conformal in terms of their periodic data. We show that if the return maps at the periodic points are, in a sense, conformal or isometric then so is the cocycle itself with respect to a Holder continuous Riemannian metric.

math.DS↗
arXiv2010

Livsic theorem for matrix cocycles

We prove the Livšic Theorem for arbitrary $GL(m,\mathbb R)$ cocycles. We consider a hyperbolic dynamical system $f : X \to X$ and a Hölder continuous function $A: X \to GL(m,\mathbb R)$. We show that if $A$ has trivial periodic data, i.e. $A(f^{n-1} p) ... A(fp) A(p) = Id$ for each periodic point $p=f^n p$, then there exists a Hölder continuous function $C: X \to GL(m,\mathbb R)$ satisfying $A (x) = C(f x) C(x) ^{-1}$ for all $x \in X$. 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 $A$.

math.DS↗
arXiv2009

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.

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arXiv2008

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 establish various properties of such systems including strong pinching, C^{1+β} 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 a torus. As a corollary we obtain local rigidity for such diffeomorphisms. We also establish a local rigidity result for Anosov diffeomorphisms in dimension three.

math.DS↗
arXiv2007

On classification of resonance-free Anosov Z^k actions

We consider actions of Z^k, k \ge 2, by Anosov diffeomorphisms which are uniformly quasiconformal on each coarse Lyapunov distribution. These actions generalize Cartan actions for which coarse Lyapunov distributions are one-dimensional. 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.

math.DS↗
arXiv2004

On the Classification of Cartan Actions

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 $k \geq 3$ whose one-parameter groups act transitively as well as nondegenerate totally nonsymplectic $\Zk$-actions for $k \geq 3$.

math.DS↗