Boundedness and invariant metrics for diffeomorphism cocycles over hyperbolic systems

Abstract

Let $\mathcal {A}$ be a Hölder continuous cocycle over a hyperbolic dynamical system with values in the group of diffeomorphisms of a compact manifold ${\mathcal {M}}$. We consider the periodic data of $\mathcal {A}$, i.e., the set of its return values along the periodic orbits in the base. We show that if the periodic data of $\mathcal {A}$ is bounded in $\text{ Diff }^{\,q}({\mathcal {M}})$, $q>1$, then the set of values of the cocycle is bounded in $\text{ Diff }^{\,r}({\mathcal {M}})$ for each $r<q$. Moreover, such a cocycle is isometric with respect to a Hölder continuous family of Riemannian metrics on ${\mathcal {M}}$.