Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-09T21:17:36.643Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-uniqueness in $\boldsymbol {G}$-measures

Published online by Cambridge University Press:  16 September 2011

A. H. DOOLEY  and
DANIEL J. RUDOLPH
A. H. DOOLEY
Affiliation:
School of Mathematics, University of New South Wales, Sydney 2052, Australia (email: a.dooley@unsw.edu.au)
DANIEL J. RUDOLPH
Affiliation:
School of Mathematics, University of New South Wales, Sydney 2052, Australia (email: a.dooley@unsw.edu.au)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Bramson and Kalikow and Quas showed the phenomenon of non-uniqueness for g-measures in the absence of a C1 condition on g. We extend this result to show that for a sequence G=(Gn), the class of G-measures can be badly behaved in the sense of containing measures of type IIIλ for all λ in a continuous image of an Fσ set.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012 , pp. 575 - 586
Copyright
Copyright © Cambridge University Press 2011

References

[1]Berger, N., Hoffman, C. and Sidoravicius, V.. Nonuniqueness for specifications in ℓ 2+ε. Preprint, available on www.arxiv.org (PR/0312344).Google Scholar
[2]Brown, G. and Dooley, A. H.. Ergodic measures are of weak product type. Math. Proc. Cambridge Philos. Soc. 98 (1985), 129–145.CrossRefGoogle Scholar
[3]Brown, G. and Dooley, A. H.. Odometer actions on G-measures. Ergod. Th. & Dynam. Sys. 11 (1991), 279–307.CrossRefGoogle Scholar
[4]Brown, G. and Dooley, A. H.. Dichotomy theorems for G-measures. Int. J. Math. 5 (1994), 827–834.CrossRefGoogle Scholar
[5]Brown, G. and Dooley, A. H.. On G-measures and product measures. Ergod. Th. & Dynam. Sys. 18 (1998), 95–107.CrossRefGoogle Scholar
[6]Brown, G., Dooley, A. H. and Lake, J.. On the Krieger–Araki–Woods ratio sets. Tohôku Univ. Math. J. 47 (1995), 1–13.Google Scholar
[7]Bramson, M. and Kalikow, S.. Nonuniqueness in g-functions. Israel J. Math. 84 (1993), 153–160.CrossRefGoogle Scholar
[8]Dooley, A. H. and Hamachi, T.. Markov odometer actions not of product type. Ergod. Th. & Dynam. Sys. 23 (2003), 1–17.CrossRefGoogle Scholar
[9]Dooley, A. H. and Hamachi, T.. Non-singular dynamical systems, Bratteli diagrams and Markov odometers. Israel J. Math. 138 (2003), 93–123.CrossRefGoogle Scholar
[10]Dooley, A. H., Klemes̆, I. and Quas, A. N.. Product and Markov measures of type III. J. Aust. Math. Soc. 64 (1988), 1–27.Google Scholar
[11]Dye, H.. On groups of measure-preserving transformations I. Amer. J. Math. 81 (1959), 119–159.CrossRefGoogle Scholar
[12]Johansson, A. and Öberg, A.. Square summability of variations and convergence of the transfer operator. Ergod. Th. & Dynam. Sys. 28 (2008), 1145–1151.CrossRefGoogle Scholar
[13]Keane, M.. Strongly mixing g-measures. Invent. Math. 16 (1972), 309–324.CrossRefGoogle Scholar
[14]Quas, A. N.. Non-ergodicity for C 1 expanding maps and g-measures. Ergod. Th. & Dynam. Sys. 16 (1966), 531–543.CrossRefGoogle Scholar