Smooth Livšic regularity for piecewise expanding maps
(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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/2224411
- author
- Nicol, Matthew and Persson, Tomas LU
- organization
- publishing date
- 2012
- 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}}, }