Uniqueness and mixing properties of Doeblin measures

被引：1

作者：

Berger, Noam ^{[1
]}

Conache, Diana ^{[1
]}

Johansson, Anders ^{[2
]}

Oberg, Anders ^{[3
]}

机构：

[1] Tech Univ Munich, Sch Computat Informat & Technol, Boltzmannstr 3, D-85748 Garching, Germany

[2] Univ Gavle, Dept Math, SE-80176 Gavle, Sweden

[3] Uppsala Univ, Dept Math, POB 480, S-75106 Uppsala, Sweden

来源：

PROBABILITY THEORY AND RELATED FIELDS | 2025年

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;

D O I：

10.1007/s00440-024-01356-3

中图分类号：

O21 [概率论与数理统计]; C8 [统计学];

学科分类号：

020208 ; 070103 ; 0714 ;

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.

引用

页数：21

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