Smooth Livšic regularity for piecewise expanding maps
type: Item
instance of: scholarly article
title: Smooth Livšic regularity for piecewise expanding maps
zbMATH Open document ID: 1263.37060
DOI: 10.1090/S0002-9939-2011-10949-3
author: Tomas Persson
author: Matthew Nicol
published in: Proceedings of the American Mathematical Society
publication date: 2012-04-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1007.4190
Mathematics Subject Classification ID: 37D50
Mathematics Subject Classification ID: 37A20
Mathematics Subject Classification ID: 37A25
Mathematics Subject Classification ID: 37E05
zbMATH DE Number: 6024097
zbMATH Keywords: cohomological equation
zbMATH Keywords: \(\beta\)-transformation
zbMATH Keywords: Gibbs-Markov map
MaRDI profile type: Publication
arXiv ID: 1007.4190
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OpenAlex ID: W2133280816
review text: \textit{A. N. Livšic} studied regularity of measurable solutions to cohomological equations [Izv. Akad. Nauk SSSR, Ser. Mat. 36, 1296--1320 (1972; Zbl 0252.58007), Math. Notes 10 (1971), 758--763 (1972; Zbl 0235.58010)]. \textit{M. Pollicott} and \textit{M. Yuri} established Livšic theorems for certain extensions of \(\beta\)-transformations [Trans. Am. Math. Soc. 351, No. 2, 559--568 (1999; Zbl 0924.58069)]. In the paper under review, the authors prove Livšic theorems for piecewise expanding maps of \([0,1)\). As part of Theorem 2, the authors show that if \(T\) is a \(\beta\)-transformation and \(\phi\) is a \(C^k\) cocycle, then bounded solutions \(\chi\) of the cohomological equation NEWLINE\[NEWLINE\phi= \chi \circ T - \chiNEWLINE\]NEWLINE have \(C^k\) versions. This extends a result of M. Pollicott and M. Yuri [loc. cit.].
reviewed by: Steve Pederson
description: scientific article; zbMATH DE number 6024097