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arXiv · 1007.4190

Smooth Livsic regularity for piecewise expanding maps

Matthew Nicol·Tomas Persson
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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}.

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Matthew Nicol, Tomas Persson. 2010-07-23. Smooth Livsic regularity for piecewise expanding maps. https://arxiv.org/abs/1007.4190

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