publication . Article . 2011

On factors of g-measures

Verbitskiy, Evgeny;
Open Access English
  • Published: 01 Dec 2011
Abstract
We show that fully supported g-measures on a shift space A(Z+), vertical bar A vertical bar <infinity, remain g-measures under single site renormalization transformations (1-block factors). (C) 2011 Royal Netherlands Academy of Arts and Sciences. Published by Elsevier B.V. All rights reserved.
Subjects
free text keywords: CONVERGENCE, SQUARE SUMMABILITY, GIBBS MEASURES, OPERATOR
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[1] Aaronson Jon, An Introduction to Infinite Ergodic Theory, in: Mathematical Surveys and Monographs, vol. 50, American Mathematical Society, Providence, RI, ISBN: 0-8218-0494-4, 1997, xii+284. MR1450400 (99d:28025).

[2] J.-R. Chazottes, E. Ugalde, Projection of Markov measures may be Gibbsian, J. Stat. Phys. (ISSN: 0022-4715) 111 (5-6) (2003) 1245-1272. doi:10.1023/A:1023056317067. MR1975928 (2004d:37008).

[3] J.-R. Chazottes, E. Ugalde, On the preservation of Gibbsianness under amalgamation of symbols, in: Papers from the Banff International Research Station Workshop on Entropy of Hidden Markov Processes and Connections to Dynamical Systems, in: B. Markus, K. Petersen, T. Weissman (Eds.), London Mathematical Society, Lecture Note Series, vol. 385, 2011, pp. 72-97.

[4] Manfred Denker, Mikhail Gordin, Gibbs measures for fibred systems, Adv. Math. (ISSN: 0001-8708) 148 (2) (1999) 161-192. MR1736956 (2001j:37061). [OpenAIRE]

[5] Ai Hua Fan, Mark Pollicott, Non-homogeneous equilibrium states and convergence speeds of averaging operators, Math. Proc. Cambridge Philos. Soc. (ISSN: 0305-0041) 129 (1) (2000) 99-115. MR1757782 (2001k:37010).

[6] Anders Johansson, Anders O¨berg, Square summability of variations of g-functions and uniqueness of g-measures, Math. Res. Lett. (ISSN: 1073-2780) 10 (5-6) (2003) 587-601. MR2024717 (2004m:37003).

[7] Anders Johansson, Anders O¨berg, Square summability of variations and convergence of the transfer operator, Ergodic Theory Dynam. Systems (ISSN: 0143-3857) 28 (4) (2008) 1145-1151. MR2437224.

[8] T. Kempton, Factors of Gibbs measures for subshifts of finite type, Bull. London Math. Soc. 43 (4) (2011) 751-764.

[9] T. Kempton, M. Pollicott, Factors of Gibbs measures for full shifts, in: Papers from the Banff International Research Station Workshop on Entropy of Hidden Markov Processes and Connections to Dynamical Systems, in: B. Markus, K. Petersen, T. Weissman (Eds.), London Mathematical Society, Lecture Note Series, vol. 385, 2011, pp. 246-257.

[10] F. Redig, F. Wang, Transformations of one-dimensional Gibbs measures with infinite range interaction, Markov Process. Related Fields 16 (4) (2010) 737-752.

[11] Aernout C.D. van Enter, Roberto Ferna´ndez, Alan D. Sokal, Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory, J. Stat. Phys. (ISSN: 0022-4715) 72 (5-6) (1993) 879-1167. MR1241537 (94m:82012).

[12] E.A. Verbitskiy, Thermodynamics of hidden Markov processes, in: Papers from the Banff International Research Station Workshop on Entropy of Hidden Markov Processes and Connections to Dynamical Systems, in: B. Markus, K. Petersen, T. Weissman (Eds.), London Mathematical Society, Lecture Note Series, vol. 385, 2011, pp. 258-272.

[13] Peter Walters, Ruelle's operator theorem and g-measures, Trans. Amer. Math. Soc. (ISSN: 0002-9947) 214 (1975) 375-387. MR0412389 (54 #515).

[14] Jisang Yoo, On factor maps that send Markov measures to Gibbs measures, J. Stat. Phys. 141 (6) (2010) 1055-1070.

Abstract
We show that fully supported g-measures on a shift space A(Z+), vertical bar A vertical bar <infinity, remain g-measures under single site renormalization transformations (1-block factors). (C) 2011 Royal Netherlands Academy of Arts and Sciences. Published by Elsevier B.V. All rights reserved.
Subjects
free text keywords: CONVERGENCE, SQUARE SUMMABILITY, GIBBS MEASURES, OPERATOR
Download from

[1] Aaronson Jon, An Introduction to Infinite Ergodic Theory, in: Mathematical Surveys and Monographs, vol. 50, American Mathematical Society, Providence, RI, ISBN: 0-8218-0494-4, 1997, xii+284. MR1450400 (99d:28025).

[2] J.-R. Chazottes, E. Ugalde, Projection of Markov measures may be Gibbsian, J. Stat. Phys. (ISSN: 0022-4715) 111 (5-6) (2003) 1245-1272. doi:10.1023/A:1023056317067. MR1975928 (2004d:37008).

[3] J.-R. Chazottes, E. Ugalde, On the preservation of Gibbsianness under amalgamation of symbols, in: Papers from the Banff International Research Station Workshop on Entropy of Hidden Markov Processes and Connections to Dynamical Systems, in: B. Markus, K. Petersen, T. Weissman (Eds.), London Mathematical Society, Lecture Note Series, vol. 385, 2011, pp. 72-97.

[4] Manfred Denker, Mikhail Gordin, Gibbs measures for fibred systems, Adv. Math. (ISSN: 0001-8708) 148 (2) (1999) 161-192. MR1736956 (2001j:37061). [OpenAIRE]

[5] Ai Hua Fan, Mark Pollicott, Non-homogeneous equilibrium states and convergence speeds of averaging operators, Math. Proc. Cambridge Philos. Soc. (ISSN: 0305-0041) 129 (1) (2000) 99-115. MR1757782 (2001k:37010).

[6] Anders Johansson, Anders O¨berg, Square summability of variations of g-functions and uniqueness of g-measures, Math. Res. Lett. (ISSN: 1073-2780) 10 (5-6) (2003) 587-601. MR2024717 (2004m:37003).

[7] Anders Johansson, Anders O¨berg, Square summability of variations and convergence of the transfer operator, Ergodic Theory Dynam. Systems (ISSN: 0143-3857) 28 (4) (2008) 1145-1151. MR2437224.

[8] T. Kempton, Factors of Gibbs measures for subshifts of finite type, Bull. London Math. Soc. 43 (4) (2011) 751-764.

[9] T. Kempton, M. Pollicott, Factors of Gibbs measures for full shifts, in: Papers from the Banff International Research Station Workshop on Entropy of Hidden Markov Processes and Connections to Dynamical Systems, in: B. Markus, K. Petersen, T. Weissman (Eds.), London Mathematical Society, Lecture Note Series, vol. 385, 2011, pp. 246-257.

[10] F. Redig, F. Wang, Transformations of one-dimensional Gibbs measures with infinite range interaction, Markov Process. Related Fields 16 (4) (2010) 737-752.

[11] Aernout C.D. van Enter, Roberto Ferna´ndez, Alan D. Sokal, Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory, J. Stat. Phys. (ISSN: 0022-4715) 72 (5-6) (1993) 879-1167. MR1241537 (94m:82012).

[12] E.A. Verbitskiy, Thermodynamics of hidden Markov processes, in: Papers from the Banff International Research Station Workshop on Entropy of Hidden Markov Processes and Connections to Dynamical Systems, in: B. Markus, K. Petersen, T. Weissman (Eds.), London Mathematical Society, Lecture Note Series, vol. 385, 2011, pp. 258-272.

[13] Peter Walters, Ruelle's operator theorem and g-measures, Trans. Amer. Math. Soc. (ISSN: 0002-9947) 214 (1975) 375-387. MR0412389 (54 #515).

[14] Jisang Yoo, On factor maps that send Markov measures to Gibbs measures, J. Stat. Phys. 141 (6) (2010) 1055-1070.

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