Cohomology of fiber bunched cocycles over hyperbolic systems

Abstract

Abstract. We consider Hölder continuous fiber bunched GL(d, R)-valued cocycles over an Anosov diffeomorphism. We show that two such cocycles are Hölder continuously cohomologous if they have equal periodic data, and prove a result for cocycles with conjugate periodic data. We obtain a corollary for cohomology between any constant cocycle and its small perturbation. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base. We show that this condition can be established based on the periodic data. Some important examples of cocycles come from the differential of the diffeomorphism and its restrictions to invariant sub-bundles. We discuss an application of our results to the question when an Anosov diffeomorphism is smoothly conjugate to a C 1 -small perturbation. We also establish Hölder continuity of a measurable conjugacy between a fiber bunched cocycle and a uniformly quasiconformal one. Our main results also hold for cocycles with values in a closed subgroup of GL(d, R), for cocycles over hyperbolic sets and shifts of finite type, and for linear cocycles on a non-trivial vector bundle. InroductionCocycles and their cohomology arise naturally in the theory of group actions and play an important role in dynamics. In this paper we study cohomology of Hölder continuous group-valued cocycles over hyperbolic dynamical systems. Our motivation comes in part from questions in local and global rigidity for hyperbolic systems and actions, where the derivative and the Jacobian provide important examples of cocycles. We state our results for the case of an Anosov diffeomorphism, but they also hold for cocycles over hyperbolic sets and symbolic dynamical systems.

Cohomology of dominated diffeomorphism-valued cocycles over hyperbolic systems

“…This generalizes the previous result independently obtained by the first author [Bac13] and Sadovskaya [Sad13] for linear cocycles (i.e. where G " GL d pRq) satisfying a fiber bunching condition.…”

Measurable rigidity of the cohomological equation for linear cocycles over hyperbolic systems

“…Our Theorem 1.4 improves on her result by removing the fiber bunching condition on A. There are examples of Hölder continuous cocycles A and B arbitrarily close to the identity which are measurably cohomologous but not continuously cohomologous that have been constructed by Pollicott and Walkden [20] (see also [22]). It's thus unclear whether a general theorem is possible in the case of Question (2).…”

“…with a α-Hölder continuous conformal structures η and P [η] which are B-invariant and A-invariant respectively. It follows from the remarks above that both A and B are uniformly quasiconformal and thus satisfy the hypotheses of Sadovskaya's theorem [22,Theorem 2.7] from which we conclude that P coincides µ-a.e. with an α-Hölder continuous function.…”

“…In this section we will show how Theorem 1.6 can be used to obtain some corollaries of independent interest which we can use to prove Theorems 1.1 and 1.4. We begin by using Theorem 1.6 to remove the fiber bunching hypothesis from one of the main theorems of a recent work of Sadovskaya [22]. Following Sadovskaya, a continuous linear cocycle A with generator A over f is uniformly quasiconformal if there is a constant L > 0 such that for every x ∈ Σ,…”

Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms

“…Remark 3.3. This proposition holds under a slightly weaker fiber bunching assumption [S,Proposition 4.4]: there exist θ < 1 and L such that for all x ∈ M, n ∈ N,…”