General Real Measurable Livšic Regularity via Transfer Operators

We prove a general measurable Livšic regularity theorem for real-valued cocycles over non-invertible dynamical systems using only abstract hypotheses on an associated transfer operator. As illustrative applications we derive measurable Livšic regularity results in the analytic regularity class for cocycles over real-analytic expanding maps, in the bounded-variation regularity class for β β -transformations, and in C^α C α regularity over the class of virtually expanding maps recently introduced by M. Tsujii.