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

Abstract

AbstractThe celebrated Livšic theorem [A. N. Livšic, Certain properties of the homology ofY-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 manifoldM, a Lie groupG, a transitive Anosov diffeomorphismfonMand a Hölder functionη:M↦Gwhose range is sufficiently close to the identity, it is sufficient for the existence of ϕ:M↦Gsatisfyingη(x)=ϕ(f(x))ϕ(x)−1that a condition—obviously necessary—on the cocycle generated byηrestricted 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.