Livsic theorems and the stable ergodicity of compact group extensions for systems with some hyperbolicity.

We consider Livsic regularity for Lie group valued cocycles over: a class of piecewise expanding maps of the interval, namely Lasota-Yorke maps; uniformly hyperbolic toral maps with singularities and a class of nonuniformly expanding interval maps. As applications of the results we prove stable ergodicity theorems for compact Lie group extension of Lasota-Yorke maps and uniformly hyperbolic toral maps with singularities. Additionally we consider conditions for the ergodicity and weak-mixing of finite group extensions of hyperbolic basic sets given in terms of periodic data and cohomological equations. We also consider stable ergodicity results for a class of nonconnected compact Lie group extensions of hyperbolic basic sets.