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November  2019, 39(11): 6277-6298. doi: 10.3934/dcds.2019274

Thermodynamic formalism for topological Markov chains on standard Borel spaces

1. 

Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, Brazil

2. 

Departamento de Matemática, Universidade Federal do Rio de Janeiro, 21941-909, Rio de Janeiro (RJ), Brazil

* Corresponding author: L. Cioletti

Received  October 2018 Revised  May 2019 Published  August 2019

Fund Project: M. Stadlbauer is supported by FAPERJ and CNPq and L. Cioletti is supported by CNPq.

We develop a Thermodynamic Formalism for bounded continuous potentials defined on the sequence space $ X\equiv E^{\mathbb{N}} $, where $ E $ is a general standard Borel space. In particular, we introduce meaningful concepts of entropy and pressure for shifts acting on $ X $ and obtain the existence of equilibrium states as finitely additive probability measures for any bounded continuous potential. Furthermore, we establish convexity and other structural properties of the set of equilibrium states, prove a version of the Perron-Frobenius-Ruelle theorem under additional assumptions on the regularity of the potential and show that the Yosida-Hewitt decomposition of these equilibrium states does not have a purely finite additive part.

We then apply our results to the construction of invariant measures of time-homogeneous Markov chains taking values on a general Borel standard space and obtain exponential asymptotic stability for a class of Markov operators. We also construct conformal measures for an infinite collection of interacting random paths which are associated to a potential depending on infinitely many coordinates. Under an additional differentiability hypothesis, we show how this process is related after a proper scaling limit to a certain infinite-dimensional diffusion.

Citation: L. Cioletti, E. Silva, M. Stadlbauer. Thermodynamic formalism for topological Markov chains on standard Borel spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6277-6298. doi: 10.3934/dcds.2019274
References:
[1]

D. AguiarL. Cioletti and R. Ruviaro, A variational principle for the specific entropy for symbolic systems with uncountable alphabets, Mathematische Nachrichten, 291 (2018), 2506-2515.  doi: 10.1002/mana.201700229.

[2]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis, A Hitchhiker's Guide, Third edition, Springer, Berlin, 2006.

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L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.

[4]

V. Baladi, Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics, 16. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633.

[5]

A. T. BaravieraL. CiolettiA. O. LopesJ. Mohr and R. R. Souza, On the general one-dimensional $XY$ model: Positive and zero temperature, selection and non-selection, Rev. Math. Phys., 23 (2011), 1063-1113.  doi: 10.1142/S0129055X11004527.

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S. Berghout, R. Fernández and E. Verbitskiy, On the relation between gibbs and $g$-measures, Ergodic Theory and Dynamical Systems, (2018), 1–26. doi: 10.1017/etds.2018.13.

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M. Bessa and M. Stadlbauer, On the Lyapunov spectrum of relative transfer operators, Stoch. Dyn., 16 (2016), 1650024, 25 pp. doi: 10.1142/S0219493716500246.

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T. Bodineau and B. Helffer, The log-Sobolev inequality for unbounded spin systems, J. Funct. Anal., 166 (1999), 168-178.  doi: 10.1006/jfan.1999.3419.

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V. I. Bogachev and A. V. Kolesnikov, The Monge-Kantorovich problem: Achievements, connections, and perspectives, Russian Mathematical Surveys, 67 (2012), 785-890.  doi: 10.1070/rm2012v067n05abeh004808.

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F. Bolley, Separability and completeness for the Wasserstein distance, in Séminaire de Probabilités XLI, Lecture Notes in Math., Springer, Berlin, 1934 (2008), 371–377. doi: 10.1007/978-3-540-77913-1_17.

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H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

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J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps, Ergodic Theory Dynam. Systems, 23 (2003), 1383-1400.  doi: 10.1017/S0143385703000087.

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M. CassandroE. OlivieriA. Pellegrinotti and E. Presutti, Existence and uniqueness of DLR measures for unbounded spin systems, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 41 (1977/78), 313-334.  doi: 10.1007/BF00533602.

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L. Cioletti, A. C. D. van Enter and R. Ruviaro, The double transpose of the ruelle operator, e-print, arXiv: 1710.03841, (2017), 1–19.

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L. Cioletti and A. O. Lopes, Ruelle operator for continuous potentials and DLR-Gibbs measures, Preprint, arXiv: 1608.03881, 2016.

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L. Cioletti and A. O. Lopes, Interactions, specifications, DLR probabilities and the Ruelle operator in the one-dimensional lattice, Discrete Contin. Dyn. Syst., 37 (2017), 6139-6152.  doi: 10.3934/dcds.2017264.

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L. Cioletti and E. A. Silva, Spectral properties of the Ruelle operator on the Walters class over compact spaces, Nonlinearity, 29 (2016), 2253-2278.  doi: 10.1088/0951-7715/29/8/2253.

[19]

V. Cyr and O. Sarig, Spectral gap and transience for Ruelle operators on countable Markov shifts, Comm. Math. Phys., 292 (2009), 637-666.  doi: 10.1007/s00220-009-0891-4.

[20]

R. L. Dobrushin and E. A. Pecherski, A criterion of the uniqueness of Gibbsian fields in the noncompact case, In Probability Theory and Mathematical Statistics (Tbilisi, 1982), Lecture Notes in Math., Springer, Berlin, 1021 (1983), 97–110. doi: 10.1007/BFb0072907.

[21]

N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7. Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958.

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T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Graduate Texts in Mathematics, 272. Springer, Cham, 2015. doi: 10.1007/978-3-319-16898-2.

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A. C. D. van EnterR. Fernández and A. D. Sokal, Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory, J. Statist. Phys., 72 (1993), 879-1167.  doi: 10.1007/BF01048183.

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R. FernándezS. Gallo and G. Maillard, Regular $g$-measures are not always Gibbsian, Electron. Commun. Probab., 16 (2011), 732-740.  doi: 10.1214/ECP.v16-1681.

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J. Fritz, Gradient dynamics of infinite point systems, Ann. Probab., 15 (1987), 478-514.  doi: 10.1214/aop/1176992156.

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J. Fröhlich and B. Zegarliński, The phase transition in the discrete Gaussian chain with $1/r^2$ interaction energy, J. Statist. Phys., 63 (1991), 455-485.  doi: 10.1007/BF01029195.

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H.-O. Georgii, Gibbs Measures and Phase Transitions, Second edition, De Gruyter Studies in Mathematics, 9. Walter de Gruyter & Co., Berlin, 2011. doi: 10.1515/9783110250329.

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J. Glueck., Existence of a separating affine functional, MathOverflow. https://mathoverflow.net/q/302479.

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H. Hennion and L. Hervé, Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness, Lecture Notes in Mathematics, 1766. Springer-Verlag, Berlin, 2001. doi: 10.1007/b87874.

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R. B. Israel and R. R. Phelps, Some convexity questions arising in statistical mechanics, Math. Scand., 54 (1984), 133-156.  doi: 10.7146/math.scand.a-12048.

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O. Jenkinson, Every ergodic measure is uniquely maximizing, Discrete Contin. Dyn. Syst., 16 (2006), 383-392.  doi: 10.3934/dcds.2006.16.383.

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C. Kardaras., Finitely additive probabilities and the fundamental theorem of asset pricing, in Contemporary Quantitative Finance, Springer, Berlin, (2010), 19–34. doi: 10.1007/978-3-642-03479-4_2.

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J. D. Knowles, Measures on topological spaces, Proc. London Math. Soc., 17 (1967), 139-156.  doi: 10.1112/plms/s3-17.1.139.

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Y. KondratievY. Kozitsky and T. Pasurek, Gibbs measures of disordered lattice systems with unbounded spins, Markov Process. Related Fields, 18 (2012), 553-582. 

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R. Lang, Unendlich-dimensionale Wienerprozesse mit Wechselwirkung. Ⅱ. Die reversiblen Masse sind kanonische Gibbs-Masse, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 39 (1977), 277-299.  doi: 10.1007/BF01877496.

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T. M. Liggett, Interacting Particle Systems, Reprint of the 1985 original, Classics in Mathematics, Springer-Verlag, Berlin, 2005. doi: 10.1007/b138374.

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J. L. Lebowitz and E. Presutti, Statistical mechanics of systems of unbounded spings, Comm. Math. Phys., 50 (1976), 195-218.  doi: 10.1007/BF01609401.

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T. Leblé and S. Serfaty, Large deviation principle for empirical fields of log and Riesz gases, Invent. Math., 210 (2017), 645-757.  doi: 10.1007/s00222-017-0738-0.

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A. O. LopesJ. K. MengueJ. Mohr and R. R. Souza, Entropy and variational principle for one-dimensional lattice systems with a general a priori probability: Positive and zero temperature, Ergodic Theory Dynam. Systems, 35 (2015), 1925-1961.  doi: 10.1017/etds.2014.15.

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R. D. Mauldin and M. Urbański, Conformal iterated function systems with applications to the geometry of continued fractions, Trans. Am. Math. Soc., 351 (1999), 4995-5025.  doi: 10.1090/S0002-9947-99-02268-0.

[42]

R. D. Mauldin and M. Urbański, Gibbs states on the symbolic space over an infinite alphabet, Israel J. Math., 125 (2001), 93-130.  doi: 10.1007/BF02773377.

[43] R. D. Mauldin and M. Urbański, Graph Directed Markov Systems, Geometry and dynamics of limit sets, Cambridge Tracts in Mathematics, 148. Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511543050.
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P. Meyer-Nieberg, Banach Lattices, Universitext, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-76724-1.

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S. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Second edition, With a prologue by Peter W. Glynn, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511626630.

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H. Osada, Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials, Ann. Probab., 41 (2013), 1-49.  doi: 10.1214/11-AOP736.

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W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, (1990), 268 pp.

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K. R. Parthasarathy, Probability Measures on Metric Spaces, Reprint of the 1967 original, AMS Chelsea Publishing, Providence, RI, 2005. doi: 10.1090/chel/352.

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S. T. Rachev, Probability Metrics and the Stability of Stochastic Models, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1991.

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show all references

References:
[1]

D. AguiarL. Cioletti and R. Ruviaro, A variational principle for the specific entropy for symbolic systems with uncountable alphabets, Mathematische Nachrichten, 291 (2018), 2506-2515.  doi: 10.1002/mana.201700229.

[2]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis, A Hitchhiker's Guide, Third edition, Springer, Berlin, 2006.

[3]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.

[4]

V. Baladi, Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics, 16. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633.

[5]

A. T. BaravieraL. CiolettiA. O. LopesJ. Mohr and R. R. Souza, On the general one-dimensional $XY$ model: Positive and zero temperature, selection and non-selection, Rev. Math. Phys., 23 (2011), 1063-1113.  doi: 10.1142/S0129055X11004527.

[6]

S. Berghout, R. Fernández and E. Verbitskiy, On the relation between gibbs and $g$-measures, Ergodic Theory and Dynamical Systems, (2018), 1–26. doi: 10.1017/etds.2018.13.

[7]

M. Bessa and M. Stadlbauer, On the Lyapunov spectrum of relative transfer operators, Stoch. Dyn., 16 (2016), 1650024, 25 pp. doi: 10.1142/S0219493716500246.

[8]

T. Bodineau and B. Helffer, The log-Sobolev inequality for unbounded spin systems, J. Funct. Anal., 166 (1999), 168-178.  doi: 10.1006/jfan.1999.3419.

[9]

V. I. Bogachev and A. V. Kolesnikov, The Monge-Kantorovich problem: Achievements, connections, and perspectives, Russian Mathematical Surveys, 67 (2012), 785-890.  doi: 10.1070/rm2012v067n05abeh004808.

[10]

F. Bolley, Separability and completeness for the Wasserstein distance, in Séminaire de Probabilités XLI, Lecture Notes in Math., Springer, Berlin, 1934 (2008), 371–377. doi: 10.1007/978-3-540-77913-1_17.

[11]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Second revised edition. With a preface by David Ruelle, Edited by Jean-René Chazottes. Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008.

[12]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[13]

J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps, Ergodic Theory Dynam. Systems, 23 (2003), 1383-1400.  doi: 10.1017/S0143385703000087.

[14]

M. CassandroE. OlivieriA. Pellegrinotti and E. Presutti, Existence and uniqueness of DLR measures for unbounded spin systems, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 41 (1977/78), 313-334.  doi: 10.1007/BF00533602.

[15]

L. Cioletti, A. C. D. van Enter and R. Ruviaro, The double transpose of the ruelle operator, e-print, arXiv: 1710.03841, (2017), 1–19.

[16]

L. Cioletti and A. O. Lopes, Ruelle operator for continuous potentials and DLR-Gibbs measures, Preprint, arXiv: 1608.03881, 2016.

[17]

L. Cioletti and A. O. Lopes, Interactions, specifications, DLR probabilities and the Ruelle operator in the one-dimensional lattice, Discrete Contin. Dyn. Syst., 37 (2017), 6139-6152.  doi: 10.3934/dcds.2017264.

[18]

L. Cioletti and E. A. Silva, Spectral properties of the Ruelle operator on the Walters class over compact spaces, Nonlinearity, 29 (2016), 2253-2278.  doi: 10.1088/0951-7715/29/8/2253.

[19]

V. Cyr and O. Sarig, Spectral gap and transience for Ruelle operators on countable Markov shifts, Comm. Math. Phys., 292 (2009), 637-666.  doi: 10.1007/s00220-009-0891-4.

[20]

R. L. Dobrushin and E. A. Pecherski, A criterion of the uniqueness of Gibbsian fields in the noncompact case, In Probability Theory and Mathematical Statistics (Tbilisi, 1982), Lecture Notes in Math., Springer, Berlin, 1021 (1983), 97–110. doi: 10.1007/BFb0072907.

[21]

N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7. Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958.

[22]

T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Graduate Texts in Mathematics, 272. Springer, Cham, 2015. doi: 10.1007/978-3-319-16898-2.

[23]

A. C. D. van EnterR. Fernández and A. D. Sokal, Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory, J. Statist. Phys., 72 (1993), 879-1167.  doi: 10.1007/BF01048183.

[24]

R. FernándezS. Gallo and G. Maillard, Regular $g$-measures are not always Gibbsian, Electron. Commun. Probab., 16 (2011), 732-740.  doi: 10.1214/ECP.v16-1681.

[25]

J. Fritz, Gradient dynamics of infinite point systems, Ann. Probab., 15 (1987), 478-514.  doi: 10.1214/aop/1176992156.

[26]

J. Fröhlich and B. Zegarliński, The phase transition in the discrete Gaussian chain with $1/r^2$ interaction energy, J. Statist. Phys., 63 (1991), 455-485.  doi: 10.1007/BF01029195.

[27]

H.-O. Georgii, Gibbs Measures and Phase Transitions, Second edition, De Gruyter Studies in Mathematics, 9. Walter de Gruyter & Co., Berlin, 2011. doi: 10.1515/9783110250329.

[28]

J. Glueck., Existence of a separating affine functional, MathOverflow. https://mathoverflow.net/q/302479.

[29]

H. Hennion and L. Hervé, Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness, Lecture Notes in Mathematics, 1766. Springer-Verlag, Berlin, 2001. doi: 10.1007/b87874.

[30]

R. B. Israel and R. R. Phelps, Some convexity questions arising in statistical mechanics, Math. Scand., 54 (1984), 133-156.  doi: 10.7146/math.scand.a-12048.

[31]

O. Jenkinson, Every ergodic measure is uniquely maximizing, Discrete Contin. Dyn. Syst., 16 (2006), 383-392.  doi: 10.3934/dcds.2006.16.383.

[32]

C. Kardaras., Finitely additive probabilities and the fundamental theorem of asset pricing, in Contemporary Quantitative Finance, Springer, Berlin, (2010), 19–34. doi: 10.1007/978-3-642-03479-4_2.

[33]

J. D. Knowles, Measures on topological spaces, Proc. London Math. Soc., 17 (1967), 139-156.  doi: 10.1112/plms/s3-17.1.139.

[34]

Y. KondratievY. Kozitsky and T. Pasurek, Gibbs measures of disordered lattice systems with unbounded spins, Markov Process. Related Fields, 18 (2012), 553-582. 

[35]

R. Lang, Unendlich-dimensionale Wienerprozesse mit Wechselwirkung. Ⅰ. Existenz, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 38 (1977), 55-72.  doi: 10.1007/BF00534170.

[36]

R. Lang, Unendlich-dimensionale Wienerprozesse mit Wechselwirkung. Ⅱ. Die reversiblen Masse sind kanonische Gibbs-Masse, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 39 (1977), 277-299.  doi: 10.1007/BF01877496.

[37]

T. M. Liggett, Interacting Particle Systems, Reprint of the 1985 original, Classics in Mathematics, Springer-Verlag, Berlin, 2005. doi: 10.1007/b138374.

[38]

J. L. Lebowitz and E. Presutti, Statistical mechanics of systems of unbounded spings, Comm. Math. Phys., 50 (1976), 195-218.  doi: 10.1007/BF01609401.

[39]

T. Leblé and S. Serfaty, Large deviation principle for empirical fields of log and Riesz gases, Invent. Math., 210 (2017), 645-757.  doi: 10.1007/s00222-017-0738-0.

[40]

A. O. LopesJ. K. MengueJ. Mohr and R. R. Souza, Entropy and variational principle for one-dimensional lattice systems with a general a priori probability: Positive and zero temperature, Ergodic Theory Dynam. Systems, 35 (2015), 1925-1961.  doi: 10.1017/etds.2014.15.

[41]

R. D. Mauldin and M. Urbański, Conformal iterated function systems with applications to the geometry of continued fractions, Trans. Am. Math. Soc., 351 (1999), 4995-5025.  doi: 10.1090/S0002-9947-99-02268-0.

[42]

R. D. Mauldin and M. Urbański, Gibbs states on the symbolic space over an infinite alphabet, Israel J. Math., 125 (2001), 93-130.  doi: 10.1007/BF02773377.

[43] R. D. Mauldin and M. Urbański, Graph Directed Markov Systems, Geometry and dynamics of limit sets, Cambridge Tracts in Mathematics, 148. Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511543050.
[44]

P. Meyer-Nieberg, Banach Lattices, Universitext, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-76724-1.

[45]

S. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Second edition, With a prologue by Peter W. Glynn, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511626630.

[46]

H. Osada, Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials, Ann. Probab., 41 (2013), 1-49.  doi: 10.1214/11-AOP736.

[47]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, (1990), 268 pp.

[48]

K. R. Parthasarathy, Probability Measures on Metric Spaces, Reprint of the 1967 original, AMS Chelsea Publishing, Providence, RI, 2005. doi: 10.1090/chel/352.

[49]

S. T. Rachev, Probability Metrics and the Stability of Stochastic Models, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1991.

[50]

D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Comm. Math. Phys., 9 (1968), 267-278.  doi: 10.1007/BF01654281.

[51]

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Figure 1.  Relations among the three theories
Figure 2.  An example of a random path $ \gamma(t) $ constructed from $ q_1,q_2,\ldots $
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