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Livsic theorems for connected Lie groups

Authors:	M Pollicott C P Walkden

Institution:	Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, U.K. ; Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, U.K.

Abstract:	Let $\phi$ be a hyperbolic diffeomorphism on a basic set $\Lambda$ and let be a connected Lie group. Let $f : \Lambda \rightarrow G$ be Hölder. Assuming that 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 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 $\phi$ , then $f = u\phi \cdot u^{-1}$ for some Hölder . These results extend well-known theorems due to Livsic when is compact or abelian.

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