SMOOTH LIVSIC REGULARITY FOR PIECEWISE EXPANDING MAPS

We consider the regularity of measurable solutions chi to the cohomological equation Phi = chi circle T - chi, where (T, X, mu) is a dynamical system and Phi: X -> R is a C-k smooth realvalued cocycle in the setting in which T: X -> X is a piecewise C-k GibbsMarkov map, an affine beta-transformation of the unit interval or more generally a piecewise C-k uniformly expanding map of an interval. We show that under mild assumptions, bounded solutions chi possess C-k versions. In particular we show that if (T, X, mu) is a beta-transformation, then chi has a C-k version, thus improving a result of Pollicott and Yuri.