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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Smooth Livšic regularity for piecewise expanding maps

Authors: Matthew Nicol and Tomas Persson
Journal: Proc. Amer. Math. Soc. 140 (2012), 905-914
MSC (2010): Primary 37D50, 37A20; Secondary 37A25
Published electronically: July 11, 2011
MathSciNet review: 2869074
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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.

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Additional Information

Matthew Nicol
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
MR Author ID: 350236

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

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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.