Uniqueness and mixing properties of Doeblin measures

• 成果类型: Article
• 署名作者: Berger, Noam; Conache, Diana; Johansson, Anders; Oberg, Anders
• 署名单位: Technical University of Munich; University of Gavle; Uppsala University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-024-01356-3
• 发表日期: 2025-12
• 页码: 1161-1181
• 关键词: Doeblin measure; Ergodic theory; g-measure; Chains with complete connections; Transfer operator; mixing; Phase Transition; SQUARE SUMMABILITY; nonuniqueness; chains; CONVERGENCE; connections; operator

摘要

In this paper we solve two open problems in ergodic theory. We prove first that if a Doeblin function g (a g-function) satisfies lim sup(n ->infinity)var(n)log g / n(-1/2 )<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.