Smooth Livšic regularity for piecewise expanding maps

Nicol, Matthew; Persson, Tomas

doi:10.1090/S0002-9939-2011-10949-3

Smooth Livšic regularity for piecewise expanding maps

Authors: Matthew Nicol and Tomas Persson
Journal: Proc. Amer. Math. Soc. 140 (2012), 905-914
MSC (2010): Primary 37D50, 37A20; Secondary 37A25
DOI: https://doi.org/10.1090/S0002-9939-2011-10949-3
Published electronically: July 11, 2011
MathSciNet review: 2869074
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Abstract: We consider the regularity of measurable solutions $\chi$ to the cohomological equation

$\displaystyle \phi = \chi \circ T -\chi,$

where $(T,X,\mu)$ is a dynamical system and $\phi \colon X\rightarrow \mathbb{R}$ is a

smooth real-valued cocycle in the setting in which $T \colon X\rightarrow X$ is a piecewise

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

versions. In particular we show that if $(T,X,\mu)$ is a $\beta$ -transformation, then $\chi$ has a

version, thus improving a result of Pollicott and Yuri.

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