Smooth Livsic regularity for piecewise expanding maps

TL;DR

This paper studies the regularity of solutions to the cohomological equation for certain expanding maps, showing that bounded solutions have smooth versions under mild conditions, thus extending previous results.

Contribution

It proves that bounded solutions to the cohomological equation for piecewise expanding maps have $C^k$ regularity, improving known results for $eta$-transformations.

Findings

Bounded solutions have $C^k$ versions under mild assumptions.

For $eta$-transformations, solutions are $C^k$, enhancing prior results.

The results apply to piecewise $C^k$ uniformly expanding maps.

Abstract

We consider the regularity of measurable solutions $\chi$ to the cohomological equation \[ \phi = \chi \circ T -\chi, \] where $(T,X,\mu)$ is a dynamical system and $\phi \colon X\rightarrow \R$ is a $C^k$ valued cocycle in the setting in which $T \colon X\rightarrow X$ is a piecewise $C^k$ Gibbs--Markov 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 et al.~\cite{Pollicott-Yuri}.