A limit theorem for the Perron-Frobenius operator of transformations on [0,1] with indifferent fixed points
type: Item
instance of: scholarly article
title: A limit theorem for the Perron-Frobenius operator of transformations on [0,1] with indifferent fixed points
zbMATH Open document ID: 0846.28007
DOI: 10.1007/BF02761642
author: Maximilian Thaler
published in: Israel Journal of Mathematics
publication date: 1996-09-23T00:00:00Z
review text: Let \(T\) be a transformation on \([0, 1]\) satisfying the author's conditions [in Isr. J. Math. 46, 67-96 (1983; Zbl 0528.28011)]. Here, it is shown that the Perron-Frobenius operator \(P\) associated to \(T\) satisfies \[ {1\over a_n} \sum^{n- 1}_{k= 0} P^k f\to h \int f \] for all integrable \(f\), where \(h\) denotes the invariant density and where the convergence is uniform away from parabolic points. In particular, this result implies the Darling-Kac property [see \textit{J. Aaronson}, \textit{M. Urbański} and the reviewer, Trans. Am. Math. Soc. 337, No. 2, 495-548 (1993; Zbl 0789.28010)].
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 820094
zbMATH Keywords: interval map
zbMATH Keywords: indifferent fixed point
zbMATH Keywords: pointwise dual ergodic
zbMATH Keywords: Darling-Kac set
zbMATH Keywords: Perron-Frobenius operator
reviewed by: Manfred Denker
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 820094