Smooth Livšic regularity for piecewise expanding maps
\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.].
scientific article; zbMATH DE number 6024097