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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Livsic theorems for connected Lie groups


Authors: M. Pollicott and C. P. Walkden
Journal: Trans. Amer. Math. Soc. 353 (2001), 2879-2895
MSC (2000): Primary 58F11; Secondary 58F15
Published electronically: March 12, 2001
MathSciNet review: 1828477
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Abstract:

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.


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Additional Information

M. Pollicott
Affiliation: Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, U.K.
Email: mp@ma.man.ac.uk

C. P. Walkden
Affiliation: Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, U.K.
Email: cwalkden@ma.man.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02708-8
PII: S 0002-9947(01)02708-8
Received by editor(s): January 31, 1999
Received by editor(s) in revised form: April 12, 2000
Published electronically: March 12, 2001
Article copyright: © Copyright 2001 American Mathematical Society