Remarks on Livšic' theory for nonabelian cocycles

Abstract

Let $(X,\phi)$ be a hyperbolic dynamical system and let $(G,\delta)$ be a Polish group. Motivated by Nicol and Pollicott, and then by Parry we study conditions under which two Hölder maps $f,g: X\longrightarrow G$ are Hölder cohomologous.

In the context of Nicol and Pollicott we show that if $f$ and $g$ are measurably cohomologous and the distortion of the metric $\delta $ by the cocycles defined by $f$ and $g$ is bounded in an appropriate sense, then $f$ and $g$ are Hölder cohomologous.

Two further results extend the main theorems recently presented by Parry. Under the hypothesis of bounded distortion we show that, if $f$ and $g$ give equal weight to all periodic points of $\phi $, then $f$ and $g$ are Hölder cohomologous. If the metric $\delta $ is bi-invariant, and if the skew-product $\phi _f$ defined by $f$ is topologically transitive, then conjugacy of weights implies that $g$ is Hölder conjugate to $\alpha \cdot f$ for some isometric automorphism $\alpha $ of $G$. The weaker condition that $g$-weights of periodic points are close to the identity whenever their $f$-weights are close to the identity implies that $g$ is continuously cohomologous to a homomorphic image of $f$.