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


The celebrated Livic theorem [A.N.Livšic, Certain properties of the homology of Y-systems, Mat.Zametki10 (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 Hlder function:MG →whose range is sufficiently close to the identity, it is sufficient for the existence of:MG →satisfying (x)=(f(x))(x)1 that a conditionobviously necessaryon the cocycle generated by 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. © 2009 Cambridge University Press.

Publication Title

Ergodic Theory and Dynamical Systems