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Journal of the London Mathematical Society 2000 61(1):286-300; doi:10.1112/S0024610799008418
© 2000 by London Mathematical Society
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© The London Mathematical Society

Livsic Regularity Theorems for Twisted Cocycle Equations Over Hyperbolic Systems

C. P. Walkden

Department of Mathematics, Manchester University Oxford Road, Manchester M13 9PL, cwalkden{at}ma.man.ac.uk

Received 24 August 1998. Revision received 4 February 1999.

Let {Phi} be a hyperbolic map. Cocycle equations of the form f = u{Phi}·g·{alpha}u–1 are considered, with f, g, u taking values in a compact connected Lie group G, {alpha} being an automorphism of G and f, g being Hölder continuous. When the eigenvalues of the derivative of {alpha} have modulus 1, it is proved that any measurable solution u has a Hölder continuous version. This condition on {alpha} is optimal. When f, g are Ck then u may be taken to be Ck–1+{varepsilon} for any {varepsilon} isin (0, 1).


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