Livsic Regularity Theorems for Twisted Cocycle Equations Over Hyperbolic Systems

Let be a hyperbolic map. Cocycle equations of the form f =u·g·u^–1 are considered, with f, g, u takingvalues in a compact connected Lie group G, being an automorphismof G and f, g being Hölder continuous. When the eigenvaluesof the derivative of have modulus 1, it is proved that anymeasurable solution u has a Hölder continuous version.This condition on is optimal. When f, g are C^k then u may betaken to be C^k–1+ for any (0, 1).