Smooth Livsic regularity for piecewise expanding maps

Abstract

We consider the regularity of measurable solutions $\ddot{I}‡$ to the cohomological equation \[ ϕ= χ\circ T -χ, \] where $(T,X,\hat{I}¼)$ is a dynamical system and $\ddot{I}•:X\to \mathbb{R}$ is a $C^k$ valued cocycle in the setting in which $T:X\to X$ is a piecewise $C^k$ Gibbs--Markov map, an affine ${\hat{I}}^2$-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 $\ddot{I}‡$ possess $C^k$ versions. In particular we show that if $(T,X,\hat{I}¼)$ is a ${\hat{I}}^2$-transformation then $\ddot{I}‡$ has a $C^k$ version, thus improving a result of Pollicott et al.~\cite{Pollicott-Yuri}.