Bernoulli Gibbs states for amenable group actions

Abstract

The ergodic properties of Gibbs states for shift-invariant specifications on a sequence space indexed by an amenable group are studied. For continuous specifications satisfying Dobrushin’s condition, it is shown that the unique Gibbs state is Bernoulli. In addition, the extreme Gibbs states (with respect to the usual partial ordering) of attractive continuous specifications are shown to be Bernoulli; in particular, a unique Gibbs state is Bernoulli. (This result has been obtained previously for Ising spin systems and -actions.) The non-continuous case is also considered.

Keywords:

• Gibbs state
• Bernoulli
• specification
• interaction potential
• amenable group action

MCS/CCS/AMS Classification/CR Category numbers:

• 37A15
• 37A60
• 60K35
• 82B05

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Erratum