Smooth Livšic regularity for piecewise expanding maps

We consider the regularity of measurable solutions χ to the cohomological equation φ = χ ◦ T − χ, where (T,X, μ) is a dynamical system and φ : X → R is a C valued cocycle in the setting in which T : X → X is a piecewise C Gibbs– Markov map, an affine β-transformation of the unit interval or more generally a piecewise C uniformly expanding map of an interval. We show that under mild assumptions, bounded solutions χ possess C versions. In particular we show that if (T,X, μ) is a β-transformation then χ has a C version, thus improving a result of Pollicott et al. [23].