General real measurable Livšic regularity via transfer operators

Abstract

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 ${\hat{I}}^2$-transformations, and in $C^{\hat{I}}\pm$ regularity over the class of virtually expanding maps recently introduced by M. Tsujii.