Linear cocycles over hyperbolic systems and criteria of conformality
Boris Kalinin Victoria Sadovskaya
Journal of Modern Dynamics 2010, 4(3): 419-441 doi: 10.3934/jmd.2010.4.419
In this paper, we study Hölder-continuous linear cocycles over transitive Anosov diffeomorphisms. Under various conditions of relative pinching we establish properties including existence and continuity of measurable invariant subbundles 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 Hölder-continuous Riemannian metric.
keywords: cocycle isometry. pinching hyperbolic system conformality periodic data
Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori
Boris Kalinin Anatole Katok
Journal of Modern Dynamics 2007, 1(1): 123-146 doi: 10.3934/jmd.2007.1.123
We prove that every smooth action $\a$ of $\mathbb{Z}^k,k\ge 2$, on the $(k+1)$-dimensional torus whose elements are homotopic to corresponding elements of an action $\a_0$ by hyperbolic linear maps preserves an absolutely continuous measure. This is the first known result concerning abelian groups of diffeomorphisms where existence of an invariant geometric structure is obtained from homotopy data.
    We also show that both ergodic and geometric properties of such a measure are very close to the corresponding properties of the Lebesgue measure with respect to the linear action $\a_0$.
keywords: $\mathbb{Z}^k$ actions. measure rigidity nonuniform hyperbolicity
Lyapunov exponents of cocycles over non-uniformly hyperbolic systems
Boris Kalinin Victoria Sadovskaya
Discrete & Continuous Dynamical Systems - A 2018, 38(10): 5105-5118 doi: 10.3934/dcds.2018224

We consider linear cocycles over non-uniformly hyperbolic dynamical systems. The base system is a diffeomorphism $f$ of a compact manifold $X$ preserving a hyperbolic ergodic probability measure $μ$. The cocycle $\mathcal{A}$ over $f$ is Hölder continuous and takes values in $GL(d, \mathbb{R})$ or, more generally, in the group of invertible bounded linear operators on a Banach space. For a $GL(d, \mathbb{R})$-valued cocycle $\mathcal{A}$ we prove that the Lyapunov exponents of $\mathcal{A}$ with respect to $μ$ can be approximated by the Lyapunov exponents of $\mathcal{A}$ with respect to measures on hyperbolic periodic orbits of $f$. In the infinite-dimensional setting one can define the upper and lower Lyapunov exponents of $\mathcal{A}$ with respect to $μ$, but they cannot always be approximated by the exponents of $\mathcal{A}$ on periodic orbits. We prove that they can be approximated in terms of the norms of the return values of $\mathcal{A}$ on hyperbolic periodic orbits of $f$.

keywords: Cocycles Lyapunov exponents non-uniformly hyperbolic systems hyperbolic measures periodic orbits
Normal forms for non-uniform contractions
Boris Kalinin Victoria Sadovskaya
Journal of Modern Dynamics 2017, 11(1): 341-368 doi: 10.3934/jmd.2017014

Let $f$ be a measure-preserving transformation of a Lebesgue space $(X,\mu)$ and let ${\mathscr{F}}$ be its extension to a bundle $\mathscr{E} = X \times {\mathbb{R}}^m$ by smooth fiber maps ${\mathscr{F}}_x : {\mathscr{E}}_x \to {\mathscr{E}}_{fx}$ so that the derivative of ${\mathscr{F}}$ at the zero section has negative Lyapunov exponents. We construct a measurable system of smooth coordinate changes ${\mathscr{H}}_x$ on ${\mathscr{E}}_x$ for $\mu$-a.e. $x$ so that the maps ${\mathscr{P}}_x ={\mathscr{H}}_{fx} \circ {\mathscr{F}}_x \circ {\mathscr{H}}_x^{-1}$ are sub-resonance polynomials in a finite dimensional Lie group. Our construction shows that such ${\mathscr{H}}_x$ and ${\mathscr{P}}_x$ are unique up to a sub-resonance polynomial. As a consequence, we obtain the centralizer theorem that the coordinate change $\mathscr{H}$ also conjugates any commuting extension to a polynomial extension of the same type. We apply our results to a measure-preserving diffeomorphism $f$ with a non-uniformly contracting invariant foliation $W$. We construct a measurable system of smooth coordinate changes ${\mathscr{H}}_x: W_x \to T_xW$ such that the maps ${\mathscr{H}}_{fx} \circ f \circ {\mathscr{H}}_x^{-1}$ are polynomials of sub-resonance type. Moreover, we show that for almost every leaf the coordinate changes exist at each point on the leaf and give a coherent atlas with transition maps in a finite dimensional Lie group.

keywords: Normal form contracting foliation non-uniform hyperbolicity Lyapunov exponents polynomial map homogeneous structure
Errata to "Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori" and "Uniqueness of large invariant measures for $\Zk$ actions with Cartan homotopy data"
Boris Kalinin Anatole Katok Federico Rodriguez Hertz
Journal of Modern Dynamics 2010, 4(1): 207-209 doi: 10.3934/jmd.2010.4.207

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