Smooth Livšic regularity for piecewise expanding maps
Author
Summary, in English
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.
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.
Department/s
- Dynamical systems
Publishing year
2012
Language
English
Pages
905-914
Publication/Series
Proceedings of the American Mathematical Society
Volume
140
Issue
3
Links
Document type
Journal article
Publisher
American Mathematical Society (AMS)
Topic
- Mathematics
Status
Published
Research group
- Analysis and Dynamics
- Dynamical systems
ISBN/ISSN/Other
- ISSN: 1088-6826