Livsic theorems for connected Lie groups

Abstract

Let ϕ \phi be a hyperbolic diffeomorphism on a basic set Λ \Lambda and let G G be a connected Lie group. Let f : Λ → G f : \Lambda \rightarrow G be Hölder. Assuming that f f satisfies a natural partial hyperbolicity assumption, we show that if u : Λ → G u : \Lambda \rightarrow G is a measurable solution to f = u ϕ ⋅ u − 1 f=u\phi \cdot u^{-1} a.e., then u u must in fact be Hölder. Under an additional centre bunching condition on f f , we show that if f f assigns ‘weight’ equal to the identity to each periodic orbit of ϕ \phi , then f = u ϕ ⋅ u − 1 f = u\phi \cdot u^{-1} for some Hölder u u . These results extend well-known theorems due to Livšic when G G is compact or abelian.