The Livšic periodic point theorem for non-abelian cocycles

Abstract

For hyperbolic systems and for Hölder cocycles with values in a compact metric group, we extend Livšic's periodic point characterisation of coboundaries. Here we show that two such cocycles are cohomologous when their respective ‘weights’ (of closed orbits) coincide. When it is only assumed that they are conjugate, one of the cocycles must (in general) be modified by an isomorphism (which stabilises conjugacy classes) to obtain cohomology. When the group is Lie and when a transitivity condition is satisfied, conjugacy of weights ensures that the cocycles are cohomologous with respect to a finitely extended group.