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Livšic theorems for non-commutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems

Published online by Cambridge University Press:  17 July 2009

RAFAEL DE LA LLAVE
Affiliation:
Department of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712, USA (email: llave@math.utexas.edu)
ALISTAIR WINDSOR
Affiliation:
373 Dunn Hall, Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, USA (email: awindsor@memphis.edu)

Abstract

The celebrated Livšic theorem [A. N. Livšic, Certain properties of the homology of Y-systems, Mat. Zametki 10 (1971), 555–564; A. N. Livšic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1296–1320] states that given a manifold M, a Lie group G, a transitive Anosov diffeomorphism f on M and a Hölder function η:MG whose range is sufficiently close to the identity, it is sufficient for the existence of ϕ:MG satisfying η(x)=ϕ(f(x))ϕ(x)−1 that a condition—obviously necessary—on the cocycle generated by η restricted to periodic orbits is satisfied. In this paper we present a new proof of the main result. These methods allow us to treat cocycles taking values in the group of diffeomorphisms of a compact manifold. This has applications to rigidity theory. The localization procedure we develop can be applied to obtain some new results on the existence of conformal structures for Anosov systems.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2009

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