Smooth Cocycle Rigidity For Expanding Maps, and an Application to Mostow Rigidity

. We consider the Livsic cocycle equation, with values in compact Lie groups, and dynamics given by a piecewise-smooth expanding Markov map. We prove that any measurable solution to this equation must necessarily have a smooth version. We deduce a smooth rigidity theorem for conjugacies between piecewise-conformal expanding Markov maps, and apply this to give a variation on the proof of Mostow's rigidity theorem. 1991 Mathematics Subject Classification: 58F15, 58F11, 30F40, 30F45 Section 1. Introduction. The hallmark of any kind of rigidity result is that an a priori mild condition of some sort forces an unexpectedly strong one. Perhaps the most striking result of this kind is Mostow's rigidity theorem, which states that, for certain hyperbolic manifolds, a homotopy equivalence in fact implies the much stronger geometric equivalence of isometry. Much recent interest has focused on the rigidity of solutions to cocyle equations over certain hyperbolic dynamical systems T : X ! X. The..