Livsic theorems for hyperbolic flows

We consider Hölder cocycle equations with values in certain Lie groups over a hyperbolic flow. We extend Livsic's results that measurable solutions to such equations must, in fact, be Hölder continuous.

     

Invariance principles for iterated maps that contract on average

We consider iterated function schemes that contract on average. Using a transfer operator approach, we prove a version of the almost sure invariance principle. This allows the system to be modelled by a Brownian motion, up to some error term. It follows that many classical statistical properties hold for such systems, such as the weak invariance principle and the law of the iterated logarithm.

     

Livsic Regularity Theorems for Twisted Cocycle Equations Over Hyperbolic Systems

Let be a hyperbolic map. Cocycle equations of the form f =u·g·u^–1 are considered, with f, g, u takingvalues in a compact connected Lie group G, being an automorphismof G and f, g being Hölder continuous. When the eigenvaluesof the derivative of have modulus 1, it is proved that anymeasurable solution u has a Hölder continuous version.This condition on is optimal. When f, g are C^k then u may betaken to be C^k–1+ for any (0, 1).     

Livsic theorems for connected Lie groups

Let be a hyperbolic diffeomorphism on a basic set and let be a connected Lie group. Let be Hölder. Assuming that satisfies a natural partial hyperbolicity assumption, we show that if is a measurable solution to a.e., then must in fact be Hölder. Under an additional centre bunching condition on , we show that if assigns `weight' equal to the identity to each periodic orbit of , then for some Hölder . These results extend well-known theorems due to Livsic when is compact or abelian.

     

Stable Ergodic Properties of Cocycles¶over Hyperbolic Attractors

We consider a hyperbolic flow φ ^t defined on an attracting basic set Λ. A map from the first (Čech) cohomology group of Λ into the dynamic cohomology group is constructed. This map is used to discuss the stable ergodicity and mixing of compact Lie group extensions and velocity changes of φ ^t . Received: 17 June 1998 / Accepted: 24 February 1999