JMD
Linear cocycles over hyperbolic systems and criteria of conformality
Boris Kalinin Victoria Sadovskaya
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
DCDS
Cohomology of $GL(2,\mathbb{R})$-valued cocycles over hyperbolic systems
Victoria Sadovskaya
We consider Hölder continuous $GL(2,\mathbb{R})$-valued cocycles over a transitive Anosov diffeomorphism. We give a complete classification up to Hölder cohomology of cocycles with one Lyapunov exponent and of cocycles that preserve two transverse Hölder continuous sub-bundles. We prove that a measurable cohomology between two such cocycles is Hölder continuous. We also show that conjugacy of periodic data for two such cocycles does not always imply cohomology, but a slightly stronger assumption does. We describe examples that indicate that our main results do not extend to general $GL(2,\mathbb{R})$-valued cocycles.
keywords: cohomology Cocycle Anosov diffeomorphism. $GL(2 Lyapunov exponent \mathbb{R})$ classification
DCDS
Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms
Boris Kalinin Victoria Sadovskaya
We consider group-valued cocycles over a partially hyperbolic diffeomorphism which is accessible volume-preserving and center bunched. We study cocycles with values in the group of invertible continuous linear operators on a Banach space. We describe properties of holonomies for fiber bunched cocycles and establish their Hölder regularity. We also study cohomology of cocycles and its connection with holonomies. We obtain a result on regularity of a measurable conjugacy, as well as a necessary and sufficient condition for existence of a continuous conjugacy between two cocycles.
keywords: Cocycle cohomology holonomy partially hyperbolic diffeomorphism.
JMD
Normal forms for non-uniform contractions
Boris Kalinin Victoria Sadovskaya

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
DCDS
Fiber bunching and cohomology for Banach cocycles over hyperbolic systems
Victoria Sadovskaya

We consider Hölder continuous cocycles over hyperbolic dynamical systems with values in the group of invertible bounded linear operators on a Banach space. We show that two fiber bunched cocycles are Hölder continuously cohomologous if and only if they have Hölder conjugate periodic data. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base system. We show that this condition can be established based on the periodic data of a cocycle. We also establish Hölder continuity of a measurable conjugacy between a fiber bunched cocycle and one with values in a set which is compact in strong operator topology.

keywords: Cocycle cohomology fiber bunching hyperbolic system periodic point bounded operator Banach space
DCDS
On the regularity of integrable conformal structures invariant under Anosov systems
Rafael De La Llave Victoria Sadovskaya
We consider conformal structures invariant under a volume-preserving Anosov system. We show that if such a structure is in $L^p$ for sufficiently large $p$, then it is continuous.
keywords: Sobolev spaces. Anosov systems Conformal structures

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