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Smooth Livšic regularity for piecewise expanding maps

Nicol, Matthew and Persson, Tomas LU orcid (2012) In Proceedings of the American Mathematical Society 140(3). p.905-914
Abstract
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... (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)
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author
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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)
alternative location
https://arxiv.org/abs/1007.4190
date added to LUP
2016-04-01 09:57:21
date last changed
2022-03-27 03:26:41
@article{4d953cad-5882-47d4-8a30-3e0c8d33dbe7,
  abstract     = {{Abstract in Undetermined<br/>We consider the regularity of measurable solutions $ \chi$ to the cohomological equation<br/>$\displaystyle \phi = \chi \circ T -\chi, $<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}},
  doi          = {{10.1090/S0002-9939-2011-10949-3}},
  volume       = {{140}},
  year         = {{2012}},
}