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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 | References | Similar Articles | Additional Information

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 $ C^k$ smooth real-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 and Yuri.


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Additional Information

Matthew Nicol
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
Email: nicol@math.uh.edu

Tomas Persson
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ulica Śniadeckich 8, P.O. Box 21, 00-956 Warszawa, Poland
Address at time of publication: Centre for Mathematical Sciences, Lund University, Box 118, 22 100 Lund, Sweden
Email: tomasp@maths.lth.se

DOI: https://doi.org/10.1090/S0002-9939-2011-10949-3
Received by editor(s): July 23, 2010
Received by editor(s) in revised form: December 14, 2010
Published electronically: July 11, 2011
Additional Notes: The second author was supported by EC FP6 Marie Curie ToK programme CODY
Communicated by: Bryna Kra
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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