August  2020, 40(8): 4625-4652. doi: 10.3934/dcds.2020195

Ruelle operator for continuous potentials and DLR-Gibbs measures

1. 

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

2. 

Departamento de Matemática - UFRGS, 91509-900, Porto Alegre, Brazil

3. 

Departamento de Matemática - UFRJ, 21941-909, Rio de Janeiro, Brazil

* Corresponding author: Leandro Cioletti

Received  January 2019 Revised  October 2019 Published  May 2020

Fund Project: The authors would like to acknowledge financial support by CAPES, CNPq (PQ 313217/2018-1, PQ 407129/2013-8, PQ 312632/2018-5, Universal 426814/2016-9) and FAP-DF

In this work we study the Ruelle Operator associated to a continuous potential defined on a countable product of a compact metric space. We prove a generalization of Bowen's criterion for the uniqueness of the eigenmeasures and that one-sided one-dimensional DLR-Gibbs measures associated to a continuous translation invariant specifications are eigenmeasures of the transpose of the Ruelle operator. From the last claim one gets that for a continuous potential the concept of eigenprobability for the transpose of the Ruelle operator is equivalent to the concept of DLR probability.

Bounded extensions of the Ruelle operator to the Lebesgue space of integrable functions, with respect to the eigenmeasures, are studied and the problem of existence of maximal positive eigenfunctions for them is considered. One of our main results in this direction is the existence of such positive eigenfunctions for Bowen's potential in the setting of a compact and metric alphabet. We also present a version of Dobrushin's Theorem in the setting of Thermodynamic Formalism.

Citation: Leandro Cioletti, Artur O. Lopes, Manuel Stadlbauer. Ruelle operator for continuous potentials and DLR-Gibbs measures. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 4625-4652. doi: 10.3934/dcds.2020195
References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, 50, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar

[2]

J. Aaronson, M. Denker and M. Urbanski, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993) 495–548. doi: 10.1090/S0002-9947-1993-1107025-2.  Google Scholar

[3]

M. Aizenman and B. Simon, A comparison of plane rotor and Ising models, Phys. Lett. A, 76 (1980), 281-282.  doi: 10.1016/0375-9601(80)90493-4.  Google Scholar

[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.  Google Scholar

[5]

A. T. BaravieraL. M. 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.  Google Scholar

[6]

S. BerghoutR. Fernández and E. Verbitskiy, On the relation between Gibbs and $g$ -measures, Ergodic Theory Dynam. Systems, 39 (2019), 3224-3249.  doi: 10.1017/etds.2018.13.  Google Scholar

[7]

T. Bousch, La condition de Walters, Ann. Sci. École Norm. Sup. (4), 34 (2001), 287–311. doi: 10.1016/S0012-9593(00)01062-4.  Google Scholar

[8]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-77695-6.  Google Scholar

[9]

L. Cioletti, M. Denker, A. O. Lopes and M. Stadlbauer, Spectral properties of the Ruelle operator for product-type potentials on shift spaces, J. Lond. Math. Soc. (2), 95 (2017), 684–704. doi: 10.1112/jlms.12031.  Google Scholar

[10]

L. Cioletti and A. O. Lopes, Phase transitions in one-dimensional translation invariant systems: A Ruelle operator approach, J. Stat. Phys., 159 (2015), 1424-1455.  doi: 10.1007/s10955-015-1202-4.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

L. CiolettiE. A. Silva and M. Stadlbauer, Thermodynamic formalism for topological Markov chains on standard Borel spaces, Discrete Contin. Dyn. Syst., 39 (2019), 6277-6298.  doi: 10.3934/dcds.2019274.  Google Scholar

[14]

M. DenkerY. Kifer and M. Stadlbauer, Thermodynamic formalism for random countable Markov shifts, Discrete Contin. Dyn. Syst., 22 (2008), 131-164.  doi: 10.3934/dcds.2008.22.131.  Google Scholar

[15]

M. Denker and M. Urbanski, On the existence of conformal measures, Trans. Amer. Math. Soc., 328 (1991), 563-587.  doi: 10.1090/S0002-9947-1991-1014246-4.  Google Scholar

[16]

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

[17]

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

[18]

J. Ginibre, General formulation of Griffiths' inequalities, Comm. Math. Phys., 16 (1970), 310-328.  doi: 10.1007/BF01646537.  Google Scholar

[19]

J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4728-9.  Google Scholar

[20]

R. B. Griffiths, Correlations in Ising ferromagnets, J. Math. Phys., 8 (1967), 478-483.  doi: 10.1063/1.1705219.  Google Scholar

[21]

R. B. Griffiths, Correlations in Ising ferromagnets. III. A mean-field bound for binary correlations, Comm. Math. Phys., 6 (1967) 121–127. doi: 10.1007/BF01654128.  Google Scholar

[22]

M. Lin, Mixing for Markov operators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 19 (1971), 231-242.  doi: 10.1007/BF00534111.  Google Scholar

[23]

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.  Google Scholar

[24]

R. Mañé, The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces, Bol. Soc. Brasil. Mat. (N.S.), 20 (1990), 1-24.  doi: 10.1007/BF02585431.  Google Scholar

[25]

S. Muir and M. Urbański, Thermodynamic formalism for a modified shift map, Stoch. Dyn., 14 (2014), 38pp. doi: 10.1142/S0219493713500202.  Google Scholar

[26]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188 (1990), 268pp.  Google Scholar

[27]

D. Ruelle, A variational formulation of equilibrium statistical mechanics and the {G}ibbs phase rule, Comm. Math. Phys., 5 (1967), 324-329.  doi: 10.1007/BF01646446.  Google Scholar

[28]

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

[29]

D. Ruelle, The thermodynamic formalism for expanding maps, Comm. Math. Phys., 125 (1989), 239-262.  doi: 10.1007/BF01217908.  Google Scholar

[30]

O. Sarig, Thermodynamic formalism for countable Markov shifts, in Hyperbolic Dynamics, Fluctuations and Large Deviations, Proc. Sympos. Pure Math., 89, Amer. Math. Soc., Providence, RI, 2015, 81–117.  Google Scholar

[31]

J. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.  doi: 10.1070/RM1972v027n04ABEH001383.  Google Scholar

[32]

H. E. Stanley, Dependence of critical properties on dimensionality of spins, Phys. Rev. Lett., 20 (1968), 589-592.  doi: 10.1103/PhysRevLett.20.589.  Google Scholar

[33]

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.  Google Scholar

[34]

P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971.  doi: 10.2307/2373682.  Google Scholar

[35]

P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc., 236 (1978), 121-153.  doi: 10.1090/S0002-9947-1978-0466493-1.  Google Scholar

[36]

P. Walters, Convergence of the Ruelle operator for a function satisfying Bowen's condition, Trans. Amer. Math. Soc., 353 (2001), 327-347.  doi: 10.1090/S0002-9947-00-02656-8.  Google Scholar

[37]

P. Walters, A natural space of functions for the Ruelle operator theorem, Ergodic Theory Dynam. Systems, 27 (2007), 1323-1348.  doi: 10.1017/S0143385707000028.  Google Scholar

show all references

References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, 50, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar

[2]

J. Aaronson, M. Denker and M. Urbanski, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993) 495–548. doi: 10.1090/S0002-9947-1993-1107025-2.  Google Scholar

[3]

M. Aizenman and B. Simon, A comparison of plane rotor and Ising models, Phys. Lett. A, 76 (1980), 281-282.  doi: 10.1016/0375-9601(80)90493-4.  Google Scholar

[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.  Google Scholar

[5]

A. T. BaravieraL. M. 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.  Google Scholar

[6]

S. BerghoutR. Fernández and E. Verbitskiy, On the relation between Gibbs and $g$ -measures, Ergodic Theory Dynam. Systems, 39 (2019), 3224-3249.  doi: 10.1017/etds.2018.13.  Google Scholar

[7]

T. Bousch, La condition de Walters, Ann. Sci. École Norm. Sup. (4), 34 (2001), 287–311. doi: 10.1016/S0012-9593(00)01062-4.  Google Scholar

[8]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-77695-6.  Google Scholar

[9]

L. Cioletti, M. Denker, A. O. Lopes and M. Stadlbauer, Spectral properties of the Ruelle operator for product-type potentials on shift spaces, J. Lond. Math. Soc. (2), 95 (2017), 684–704. doi: 10.1112/jlms.12031.  Google Scholar

[10]

L. Cioletti and A. O. Lopes, Phase transitions in one-dimensional translation invariant systems: A Ruelle operator approach, J. Stat. Phys., 159 (2015), 1424-1455.  doi: 10.1007/s10955-015-1202-4.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

L. CiolettiE. A. Silva and M. Stadlbauer, Thermodynamic formalism for topological Markov chains on standard Borel spaces, Discrete Contin. Dyn. Syst., 39 (2019), 6277-6298.  doi: 10.3934/dcds.2019274.  Google Scholar

[14]

M. DenkerY. Kifer and M. Stadlbauer, Thermodynamic formalism for random countable Markov shifts, Discrete Contin. Dyn. Syst., 22 (2008), 131-164.  doi: 10.3934/dcds.2008.22.131.  Google Scholar

[15]

M. Denker and M. Urbanski, On the existence of conformal measures, Trans. Amer. Math. Soc., 328 (1991), 563-587.  doi: 10.1090/S0002-9947-1991-1014246-4.  Google Scholar

[16]

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

[17]

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

[18]

J. Ginibre, General formulation of Griffiths' inequalities, Comm. Math. Phys., 16 (1970), 310-328.  doi: 10.1007/BF01646537.  Google Scholar

[19]

J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4728-9.  Google Scholar

[20]

R. B. Griffiths, Correlations in Ising ferromagnets, J. Math. Phys., 8 (1967), 478-483.  doi: 10.1063/1.1705219.  Google Scholar

[21]

R. B. Griffiths, Correlations in Ising ferromagnets. III. A mean-field bound for binary correlations, Comm. Math. Phys., 6 (1967) 121–127. doi: 10.1007/BF01654128.  Google Scholar

[22]

M. Lin, Mixing for Markov operators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 19 (1971), 231-242.  doi: 10.1007/BF00534111.  Google Scholar

[23]

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.  Google Scholar

[24]

R. Mañé, The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces, Bol. Soc. Brasil. Mat. (N.S.), 20 (1990), 1-24.  doi: 10.1007/BF02585431.  Google Scholar

[25]

S. Muir and M. Urbański, Thermodynamic formalism for a modified shift map, Stoch. Dyn., 14 (2014), 38pp. doi: 10.1142/S0219493713500202.  Google Scholar

[26]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188 (1990), 268pp.  Google Scholar

[27]

D. Ruelle, A variational formulation of equilibrium statistical mechanics and the {G}ibbs phase rule, Comm. Math. Phys., 5 (1967), 324-329.  doi: 10.1007/BF01646446.  Google Scholar

[28]

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

[29]

D. Ruelle, The thermodynamic formalism for expanding maps, Comm. Math. Phys., 125 (1989), 239-262.  doi: 10.1007/BF01217908.  Google Scholar

[30]

O. Sarig, Thermodynamic formalism for countable Markov shifts, in Hyperbolic Dynamics, Fluctuations and Large Deviations, Proc. Sympos. Pure Math., 89, Amer. Math. Soc., Providence, RI, 2015, 81–117.  Google Scholar

[31]

J. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.  doi: 10.1070/RM1972v027n04ABEH001383.  Google Scholar

[32]

H. E. Stanley, Dependence of critical properties on dimensionality of spins, Phys. Rev. Lett., 20 (1968), 589-592.  doi: 10.1103/PhysRevLett.20.589.  Google Scholar

[33]

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.  Google Scholar

[34]

P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971.  doi: 10.2307/2373682.  Google Scholar

[35]

P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc., 236 (1978), 121-153.  doi: 10.1090/S0002-9947-1978-0466493-1.  Google Scholar

[36]

P. Walters, Convergence of the Ruelle operator for a function satisfying Bowen's condition, Trans. Amer. Math. Soc., 353 (2001), 327-347.  doi: 10.1090/S0002-9947-00-02656-8.  Google Scholar

[37]

P. Walters, A natural space of functions for the Ruelle operator theorem, Ergodic Theory Dynam. Systems, 27 (2007), 1323-1348.  doi: 10.1017/S0143385707000028.  Google Scholar

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