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Unique Bernoulli g-measures
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
University of Warwick, Coventry, England.
2012 (English)In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 14, no 5, 1599-1615 p.Article in journal (Refereed) Published
Abstract [en]

We improve and subsume the conditions of Johansson and O¨ berg [18] and Berbee [2]for uniqueness of a g-measure, i.e., a stationary distribution for chains with complete connections.In addition, we prove that these unique g-measures have Bernoulli natural extensions. In particular,we obtain a unique g-measure that has the Bernoulli property for the full shift on finitely manystates under any one of the following additional assumptions.

(1)P1n=1(varn log g)2 < 1,(2) For any fixed ✏ > 0,P1n=1 e−(1/2+✏)(var1 log g+···+varn log g) = 1,(3) varn log g = o(1/pn) as n!1.

That the measure is Bernoulli in the case of (1) is new. In (2) we have an improved version ofBerbee’s [2] condition (concerning uniqueness and Bernoullicity), allowing the variations of log gto be essentially twice as large. Finally, (3) is an example that our main result is new both foruniqueness and for the Bernoulli property.We also conclude that we have convergence in the Wasserstein metric of the iterates of theadjoint transfer operator to the g-measure.

Place, publisher, year, edition, pages
European Mathematical Society Publishing House, 2012. Vol. 14, no 5, 1599-1615 p.
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-126899DOI: 10.4171/JEMS/342ISI: 000308126300009OAI: oai:DiVA.org:uu-126899DiVA: diva2:327596
Available from: 2010-06-30 Created: 2010-06-29 Last updated: 2017-12-12Bibliographically approved

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