Smooth Livšic regularity for piecewise expanding maps
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- by Matthew Nicol and Tomas Persson PDF
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Abstract:
We consider the regularity of measurable solutions $\chi$ to the cohomological equation \[ \phi = \chi \circ T -\chi , \] 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.References
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Additional Information
- Matthew Nicol
- Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
- MR Author ID: 350236
- 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
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2869074