publication . Preprint . 2005

Countable state shifts and uniqueness of g-measures

Johansson, Anders; Öberg, Anders; Pollicott, Mark;
Open Access English
  • Published: 05 Sep 2005
Abstract
In this paper we present a new approach to studying g-measures which is based upon local absolute continuity. We extend the result in [11] that square summability of variations of g-functions ensures uniqueness of g-measures. The first extension is to the case of countably many symbols. The second extension is to some cases where $g \geq 0$, relaxing the earlier requirement in [11] that inf g>0.
Subjects
free text keywords: 37A05, 28D05, 37A30, 37A60, 82B20, Mathematics - Probability, Mathematics - Dynamical Systems
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19 references, page 1 of 2

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

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

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

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

[5] H.J. Engelbert and A.N. Shiryaev, On absolute continuity and singularity of probability measures, Banach Cent. Publ. 6 (1980), 121-132.

[6] R. Fernandez and G. Maillard, Chains with Complete Connections: General Theory, Uniqueness, Loss of Memory and Mixing Properties, J. Statist. Phys. 118 (2005), no. 3-4, 555-588.

[7] P. Hulse Ph.D. Thesis , Warwick University (1980).

[8] J. Jacod and A.N. Shiryaev, Limit Theorems for Stochastic Processes, 2nd ed., Grundlehren der mathematischen Wissenschaften 288, Springer-Verlag 2003.

[9] Yu. Kabanov, R.S. Lipster and A.N. Siryaev, On the question of the absolute continuity and singularity of probability measures (Russian), Mat. Sb. (N.S.) 104(146) (1977), no. 2(10), 227-247.

[10] M. Keane, Strongly Mixing g-Measures, Invent. Math. 16 (1972), 309-324.

[11] A. Johansson and A. O¨berg, Square summability of variations of g-functions and uniqueness of g-measures, Math. Res. Lett. 10 (2003), no. 5-6, 587-601.

[12] D. Mauldin and M. Urbanski, Gibbs states on the symbolic space over an infinite alphabet, Israel J. Math. 125 (2001), 93-130.

[13] O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems 19 (1999), no. 6, 1565-1593.

[14] O. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math. 121 (2001), 285-311.

[15] O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1751-1758. [OpenAIRE]

19 references, page 1 of 2
Abstract
In this paper we present a new approach to studying g-measures which is based upon local absolute continuity. We extend the result in [11] that square summability of variations of g-functions ensures uniqueness of g-measures. The first extension is to the case of countably many symbols. The second extension is to some cases where $g \geq 0$, relaxing the earlier requirement in [11] that inf g>0.
Subjects
free text keywords: 37A05, 28D05, 37A30, 37A60, 82B20, Mathematics - Probability, Mathematics - Dynamical Systems
Download from
19 references, page 1 of 2

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

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

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

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

[5] H.J. Engelbert and A.N. Shiryaev, On absolute continuity and singularity of probability measures, Banach Cent. Publ. 6 (1980), 121-132.

[6] R. Fernandez and G. Maillard, Chains with Complete Connections: General Theory, Uniqueness, Loss of Memory and Mixing Properties, J. Statist. Phys. 118 (2005), no. 3-4, 555-588.

[7] P. Hulse Ph.D. Thesis , Warwick University (1980).

[8] J. Jacod and A.N. Shiryaev, Limit Theorems for Stochastic Processes, 2nd ed., Grundlehren der mathematischen Wissenschaften 288, Springer-Verlag 2003.

[9] Yu. Kabanov, R.S. Lipster and A.N. Siryaev, On the question of the absolute continuity and singularity of probability measures (Russian), Mat. Sb. (N.S.) 104(146) (1977), no. 2(10), 227-247.

[10] M. Keane, Strongly Mixing g-Measures, Invent. Math. 16 (1972), 309-324.

[11] A. Johansson and A. O¨berg, Square summability of variations of g-functions and uniqueness of g-measures, Math. Res. Lett. 10 (2003), no. 5-6, 587-601.

[12] D. Mauldin and M. Urbanski, Gibbs states on the symbolic space over an infinite alphabet, Israel J. Math. 125 (2001), 93-130.

[13] O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems 19 (1999), no. 6, 1565-1593.

[14] O. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math. 121 (2001), 285-311.

[15] O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1751-1758. [OpenAIRE]

19 references, page 1 of 2
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