# Livsic theorems for connected Lie groups

Pollicott, M. and Walkden, C. P. (2001) Livsic theorems for connected Lie groups. Transactions of the American Mathematical Society, 353 (7). pp. 2879-2895. ISSN 1088-6850

Let $\phi$ be a hyperbolic diffeomorphism on a basic set $\Lambda$ and let $G$ be a connected Lie group. Let $f : \Lambda \rightarrow G$ be Hölder. Assuming that $f$ satisfies a natural partial hyperbolicity assumption, we show that if $u : \Lambda \rightarrow G$ is a measurable solution to $f=u\phi \cdot u^{-1}$ a.e., then $u$ must in fact be Hölder. Under an additional centre bunching condition on $f$, we show that if $f$ assigns `weight' equal to the identity to each periodic orbit of $\phi$, then $f = u\phi \cdot u^{-1}$ for some Hölder $u$. These results extend well-known theorems due to Livsic when $G$ is compact or abelian.