Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T00:59:40.298Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-uniqueness in -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: [email protected])
DANIEL J. RUDOLPH
Affiliation:
School of Mathematics, University of New South Wales, Sydney 2052, Australia (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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