Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-01-07T19:31:23.759Z Has data issue: false hasContentIssue false
  • English
  • Français

On the equivalence of certain ergodic properties for Gibbs states

Published online by Cambridge University Press:  01 February 2000

FRANK DEN HOLLANDER  and
JEFFREY E. STEIF
FRANK DEN HOLLANDER
Affiliation:
Department of Mathematics, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands (e-mail: denholla@sci.kun.nl)
JEFFREY E. STEIF
Affiliation:
Department of Mathematics, Chalmers University of Technology, S–41296 Gothenburg, Sweden (e-mail: steif@math.chalmers.se) Current address: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend our previous work by proving that for translation invariant Gibbs states on ${\mathbb Z}^d$ with a translation invariant interaction potential $\Psi=(\Psi_A)$ satisfying $\sum_{A \ni 0}|A|^{-1}[\diam(A)]^d\|\Psi_A\|<\infty$ the following hold: (1) the Kolmogorov-property implies a trivial full tail and (2) the Bernoulli-property implies Følner independence. The existence of bilaterally deterministic Bernoulli Shifts tells us that neither (1) nor (2) is, in general, true for random fields without some further assumption (even when $d=1$).

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 20 , Issue 1 , February 2000 , pp. 231 - 239
Copyright
2000 Cambridge University Press

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.)