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Square summability of variations and convergence of the transfer operator

Published online by Cambridge University Press:  01 August 2008

ANDERS JOHANSSON  and
ANDERS ÖBERG
ANDERS JOHANSSON
Affiliation:
Sub-department of Mathematics and Statistics, University of Gävle, SE-801 76 Gävle, Sweden (email: ajj@hig.se)
ANDERS ÖBERG
Affiliation:
Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden (email: anders@math.uu.se)
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Abstract

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In this paper we study the one-sided shift operator on a state space defined by a finite alphabet. Using a scheme developed by Walters [P. Walters. Trans. Amer. Math. Soc.353(1) (2001), 327–347], we prove that the sequence of iterates of the transfer operator converges under square summability of variations of the g-function, a condition which gave uniqueness of a g-measure in our earlier work [A. Johansson and A. Öberg. Math. Res. Lett.10(5–6) (2003), 587–601]. We also prove uniqueness of the so-called G-measures, introduced by Brown and Dooley [G. Brown and A. H. Dooley. Ergod. Th. & Dynam. Sys.11 (1991), 279–307], under square summability of variations.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 4 , August 2008 , pp. 1145 - 1151
Copyright
Copyright © 2008 Cambridge University Press

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