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Smooth Livšic regularity for piecewise expanding maps

Nicol, Matthew and Persson, Tomas LU (2012) In Proceedings of the American Mathematical Society 140(3). p.905-914
Abstract (Swedish)
Abstract in Undetermined

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... (More)
Abstract in Undetermined

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. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Proceedings of the American Mathematical Society
volume
140
issue
3
pages
905 - 914
publisher
American Mathematical Society (AMS)
external identifiers
  • scopus:82255164047
ISSN
1088-6826
DOI
10.1090/S0002-9939-2011-10949-3
language
English
LU publication?
yes
id
4d953cad-5882-47d4-8a30-3e0c8d33dbe7 (old id 2224411)
date added to LUP
2012-01-27 15:21:02
date last changed
2017-03-14 13:40:53
@article{4d953cad-5882-47d4-8a30-3e0c8d33dbe7,
  abstract     = {<b>Abstract in Undetermined</b><br/><br>
We consider the regularity of measurable solutions $ \chi$ to the cohomological equation <br/><br>
$\displaystyle \phi = \chi \circ T -\chi, $ <br/><br>
 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.},
  author       = {Nicol, Matthew and Persson, Tomas},
  issn         = {1088-6826},
  language     = {eng},
  number       = {3},
  pages        = {905--914},
  publisher    = {American Mathematical Society (AMS)},
  series       = {Proceedings of the American Mathematical Society},
  title        = {Smooth Livšic regularity for piecewise expanding maps},
  url          = {http://dx.doi.org/10.1090/S0002-9939-2011-10949-3},
  volume       = {140},
  year         = {2012},
}