publication . Preprint . Article . 2010

Unique Bernoulli $g$-measures

Anders Johannson; Anders Öberg; Mark Pollicott;
Open Access English
  • Published: 05 Apr 2010
We improve and subsume the conditions of Johansson and \"Oberg [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 many states under any one of the following additional assumptions. (1) $$\sum_{n=1}^\infty (\var_n \log g)^2<\infty,$$ (2) For any fixed $\epsilon>0$, $$\sum_{n=1}^\infty e^{-(\{1}{2}+\epsilon) (\var_1 \log g+...+\var_n \log g)}=\infty,$$ (3) $$\var_n \log g=\ordo{\{1}{\sqrt{n}}}, \quad n\to ...
free text keywords: Mathematics - Dynamical Systems, Mathematics - Probability, Primary 37A05, 37A35, 60G10, Applied Mathematics, General Mathematics, Transfer operator, Uniqueness, Topology, Bernoulli's principle, Mathematical analysis, Iterated function, Wasserstein metric, Bernoulli process, Bernoulli scheme, Mathematics, Stationary distribution
28 references, page 1 of 2

[1] M. Aizenman, J.T. Chayes, L. Chayes, C.M. Newman, Discontinuity of the magnetization in one-dimensional 1/|x − y|2 Ising and Potts models, J. Statist. Phys. 50 (1988), no. 1-2, 1-40. [OpenAIRE]

[2] H. Berbee, Chains with Infinite Connections: Uniqueness and Markov Representation, Probab. Theory Related Fields 76 (1987), 243-253. [OpenAIRE]

[3] H. Berbee, Uniqueness of Gibbs measures and absorption probabilities, Ann. Probab. 17 (1989), no. 4, 1416-1431. [OpenAIRE]

[4] N. Berger, C. Hoffman and V. Sidoravicius, Nonuniqueness for specifications in l2+ǫ . Preprint available on (PR/0312344).

[5] M. Bramson and S. Kalikow, Nonuniqueness in g-functions, Israel J. Math. 84 (1993), 153-160.

[6] X. Bressaud, R. Ferna´ndez and A. Galves, Decay of correlations for non H¨olderian dynamics. A coupling approach, Electron. J. Proabab. 4 (1999), no. 3, 1-19.

[7] Z. Coelho and A. Quas, Criteria for d¯-continuity, Trans. Amer. Math. Soc. 350 (1998), no. 8, 3257-3268.

[8] P. Diaconis and D. Freedman, Iterated Random Functions, SIAM Review 41 (1999), no. 1, 45-76.

[9] W. Doeblin and R. Fortet, Sur des chaˆınes `a liaisons compl`etes, Bull. Soc. Math. France 65 (1937), 132-148.

[10] F.J. Dyson, Non-existence of spontaneous magnetisation in a one-dimensional Ising ferromagnet, Commun. Math. Phys. 12 (1969), no. 3, 212-215. [OpenAIRE]

[11] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, John Wiley & Sons, 1971.

[12] T.E. Harris, On chains of infinite order, Pacific J. Math. 5 (1955), 707-724.

[13] P. Hulse, An example of non-unique g-measures, Ergodic Theory Dynam. Systems 26 (2006), no. 2, 439-445. [OpenAIRE]

[14] M. Iosifescu, Iterated function sytems. A critical survey, Math. Rep. (Bucur.) 11(61) (2009), no. 3, 181-229.

[15] M. Iosifescu and S. Grigorescu, Dependence with complete connections and its applications, Cambridge University Press, 1990.

28 references, page 1 of 2
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publication . Preprint . Article . 2010

Unique Bernoulli $g$-measures

Anders Johannson; Anders Öberg; Mark Pollicott;