Doeblin measures: uniqueness and mixing properties

2023/03/24 by Berger, Noam, Conache, Diana, Johannson, Anders +1 · 1 citation
#37A05 #37A25 #37A50 #60G10 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2303.13891

Abstract

In this paper we solve two open problems in ergodic theory. We prove first that if a Doeblin function g (a g-function) satisfies \limsup_n→∞\fracvar_n log gn^-1/2 lt; 2, then we have a unique Doeblin measure (g-measure). This result indicates a possible phase transition in analogy with the long-range Ising model. Secondly, we provide an example of a Doeblin function with a unique Doeblin measure that is not weakly mixing, which implies that the sequence of iterates of the transfer operator does not converge, solving a well-known folklore problem in ergodic theory. Previously it was only known that uniqueness does not imply the Bernoulli property.

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• Unique Bernoulli Gibbs states and g-measures

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