Skip to main content
SpringerLink
Log in

The double transpose of the Ruelle operator

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

In this paper we study the double transpose of the \(L^1(X,\mathscr {B}(X),\nu )\)-extensions of the Ruelle transfer operator \(\mathscr {L}_{f}\) associated to a general real continuous potential \(f\in C(X)\), where \(X=E^{\mathbb {N}}\), the alphabet E is any compact metric space and \(\nu \) is a maximal eigenmeasure. For this operator, denoted by \(\mathbb {L}^{**}_{f}\), we prove the existence of some non-negative eigenfunction, in the Banach lattice sense, associated to \(\rho (\mathscr {L}_{f})\), the spectral radius of the Ruelle operator acting on C(X). As an application, we obtain a sufficient condition ensuring that the extension of the Ruelle operator to \(L^1(X,\mathscr {B}(X),\nu )\) has an eigenfunction associated to \(\rho (\mathscr {L}_{f})\). These eigenfunctions agree with the usual maximal eigenfunctions, when the potential f belongs to the Hölder, Walters or Bowen class. We also construct solutions to the classical and generalized variational problem, using the eigenvector constructed here.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aaronson, J., Denker, M., Urbański, M.: Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Am. Math. Soc. 337(2), 495–548 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adler, R.L., Weiss, B.: Entropy, a complete metric invariant for automorphisms of the torus. Proc. Nat. Acad. Sci. USA 57, 1573–1576 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aguiar, D., Cioletti, L., Ruviaro, R.: A variational principle for the specific entropy for symbolic systems with uncountable alphabets. Math. Nachr. 291(17–18), 2506–2515 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis, III Springer, Berlin (2006)

    MATH  Google Scholar 

  5. Baladi, V.: Positive transfer operators and decay of correlations. In: Advanced Series in Nonlinear Dynamics, vol. 16. World Scientific Publishing Co. Inc, River Edge (2000)

  6. Baraviera, A.T., Cioletti, L., Lopes, A.O., Mohr, J., Souza, R.R.: On the general one-dimensional \(XY\) model: positive and zero temperature, selection and non-selection. Rev. Math. Phys. 23(10), 1063–1113 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Barreira, L.: Thermodynamic formalism and applications to dimension theory. In: Progress in Mathematics, vol. 294. Birkhäuser, Basel (2011)

  8. Berger, N., Hoffman, C., Sidoravicius, V.: Non-uniqueness for specifications in \(\ell ^{2+\varepsilon }\). In: Ergodic Theory and Dynamical Systems, pp. 1–11 (2017)

  9. Bis, A., Carvalho, M., Mendes, M., Varandas, P.: A convex analysis approach to entropy functions, variational principles and equilibrium states. Commun. Math. Phys. 394, 215–256 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bishop, E., Phelps, R.R.: The support functionals of a convex set. In: Proceedings of Symposium on Pure Mathematics, vol. VII. Amer. Math. Soc., Providence, pp. 27–35 (1963)

  11. Bousch. La condition de Walters. Ann. Sci. École Norm. Sup. (4), 34(2), 287–311 (2001)

  12. Bowen, R.: Markov partitions for axiom \({\rm A}\) diffeomorphisms. Am. J. Math. 92, 725–747 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bowen, R.: Bernoulli equilibrium states for axiom A diffeomorphisms. Math. Syst. Theory, 8(4), 289–294 (1974/1975)

  14. Bowen, R.: Hausdorff dimension of quasicircles. Inst. Hautes Études Sci. Publ. Math. 50(1), 11–25 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bowen, R.: In: Chazottes, J.R. (ed.), Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, revised edition, vol. 470. Springer, Berlin (2008)

  16. Bramson, M., Kalikow, S.: Nonuniqueness in \(g\)-functions. Isr. J. Math. 84(1–2), 153–160 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Cioletti, L., Lopes, A., Stadlbauer, M.: Ruelle operator for continuous potentials and DLR-Gibbs measures. Discrete Contin. Dyn. Syst. Ser. A 40(8), 4625–4652 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cioletti, L., Lopes, A.O.: Phase transitions in one-dimensional translation invariant systems: a Ruelle operator approach. J. Stat. Phys. 159(6), 1424–1455 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cioletti, L., Melo, L., Ruviaro, R., Silva, E.A.: On the dimension of the space of harmonic functions on transitive shift spaces. Adv. Math. 385, 107758 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cioletti, L., Silva, E.A.: Spectral properties of the Ruelle operator on the Walters class over compact spaces. Nonlinearity 29(8), 2253–2278 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Daniëls, H.A.M., van Enter, A.C.D.: Differentiability properties of the pressure in lattice systems. Commun. Math. Phys. 71(1), 65–76 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  23. Denker, M., Urbański, M.: On the existence of conformal measures. Trans. Am. Math. Soc. 328(2), 563–587 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dunford, N., Schwartz, J.T.: Linear operators. I: general theory with the assistance of W. G. Bade and R. G. Bartle. In: Pure and Applied Mathematics, vol. 7. Interscience Publishers, Inc., New York (1958)

  25. Dyson, F.J.: Existence of a phase-transition in a one-dimensional Ising ferromagnet. Commun. Math. Phys. 12(2), 91–107 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fan, A., Jiang, Y.: On Ruelle–Perron–Frobenius operators, I: Ruelle theorem. Commun. Math. Phys. 223(1), 125–141 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Fan, A., Jiang, Y.: On Ruelle–Perron–Frobenius operators, II: convergence speeds. Commun. Math. Phys. 223(1), 143–159 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Fisher, M.E., Felderhof, B.U.: Phase transitions in one-dimensional cluster-interaction fluids IA. Thermodyn. Ann. Phys. 58, 176–216 (1970)

    Article  Google Scholar 

  29. Georgii, H.O.: Gibbs Measures and Phase Transitions, de Gruyter Studies in Mathematics, 2nd edn., vol. 9. Walter de Gruyter & Co., Berlin (2011)

  30. Gurevich, B.M., Savchenko, S.V.: Thermodynamic formalism for symbolic Markov chains with a countable number of states. Uspekhi Mat. Nauk 53, 3–106 (1998)

    MathSciNet  Google Scholar 

  31. Hofbauer, F.: Examples for the nonuniqueness of the equilibrium state. Trans. Am. Math. Soc. 228, 223–241 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  32. Israel, R.B.: Existence of phase transitions for long-range interactions. Commun. Math. Phys. 43, 59–68 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  33. Israel, R.B.: Convexity in the Theory of Lattice Gases: Princeton Series in Physics with an introduction by Arthur S, Wightman, Princeton University Press, Princeton (1979)

  34. Israel, R.B., Phelps, R.R.: Some convexity questions arising in statistical mechanics. Math. Scand. 54(1), 133–156 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  35. Jenkinson, O.: Every ergodic measure is uniquely maximizing. Discrete Contin. Dyn. Syst. 16(2), 383–392 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Johansson, A., Öberg, A., Pollicott, M.: Phase transitions in long-range Ising models and an optimal condition for factors of \(g\)-measures. Ergod. Theory Dyn. Syst. 39(5), 1317–1330 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  37. Keller, G.: Equilibrium States in Ergodic Theory. London Mathematical Society Student Texts, vol. 42. Cambridge University Press, Cambridge (1998)

  38. Kloeckner, B.: An optimal transportation approach to the decay of correlations for non-uniformly expanding maps. Ergod. Theory Dyn. Syst. 40(3), 714–750 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kramers, H.A., Wannier, G.H.: Statistics of the two-dimensional ferromagnet. I. Phys. Rev. 2(60), 252–262 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  40. Ledrappier, F.: Principe variationnel et systèmes dynamiques symboliques. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 30, 185–202 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  41. Liverani, C., Saussol, B., Vaienti, S.: A probabilistic approach to intermittency. Ergod. Theory Dyn. Syst. 19(3), 671–685 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  42. Lopes, A.O., Mengue, J.K., Mohr, J., Souza, R.R.: Entropy and variational principle for one-dimensional lattice systems with a general a priori probability: positive and zero temperature. Ergod. Theory Dyn. Syst. 35(6), 1925–1961 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  43. Mañé, R.: The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces. Bol. Soc. Brasil. Mat. 20(2), 1–24 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  44. Mauldin, R.D., Urbański, M.: Gibbs states on the symbolic space over an infinite alphabet. Isr. J. Math. 125, 93–130 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  45. Mayer, D.H.: The Ruelle–Araki Transfer Operator in Classical Statistical Mechanics. Lecture Notes in Physics, vol. 123. Springer, Berlin (1980)

  46. McCluskey, H., Manning, A.: Hausdorff dimension for horseshoes. Ergod. Theory Dyn. Syst. 3(2), 251–260 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  47. Montroll, E.W.: Statistical mechanics of nearest neighbor systems. J. Chem. Phys. 9(9), 706–721 (1941)

    Article  Google Scholar 

  48. Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187(188), 268 (1990)

    MathSciNet  MATH  Google Scholar 

  49. Pollicott, M.: Maximal Lyapunov exponents for random matrix products. Invent. Math. 181(1), 209–226 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  50. Prellberg, T., Slawny, J.: Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions. J. Stat. Phys. 66(1–2), 503–514 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  51. Ruelle, D.: A variational formulation of equilibrium statistical mechanics and the Gibbs phase rule. Commun. Math. Phys. 5, 324–329 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  52. Ruelle, D.: Statistical mechanics of a one-dimensional lattice gas. Commun. Math. Phys. 9, 267–278 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  53. Ruelle, D.: Dynamical zeta functions and transfer operators. Notices Am. Math. Soc. 49(8), 887–895 (2002)

    MathSciNet  MATH  Google Scholar 

  54. Ruelle, D.: Thermodynamic Formalism. Cambridge Mathematical Library, The Mathematical Structures of Equilibrium Statistical Mechanics, 2nd edn. Cambridge University Press, Cambridge (2004)

  55. Sarig, O.: Thermodynamic formalism for countable Markov shifts. Ergod. Theory Dyn. Syst. 19(6), 1565–1593 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  56. Sarig, O.: Existence of Gibbs measures for countable Markov shifts. Proc. Am. Math. Soc. 131(6), 1751–1758 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  57. Sewell, G.L.: Metastable states of quantum lattice systems. Commun. Math. Phys. 55(1), 63–66 (1977)

    Article  MathSciNet  Google Scholar 

  58. Silva, E.A.: Pressure Derivative on Uncountable Alphabet Setting: A Ruelle Operator Approach. arXiv:1707.09072 (2017)

  59. Sinaĭ, J.G.: Gibbs measures in ergodic theory. Uspehi Mat. Nauk 27, 21–64 (1972)

    MathSciNet  MATH  Google Scholar 

  60. Sinaĭ, J.G.: Markov partitions and C-diffeomorphisms. Funkcional. Anal. i Priložen 2(1), 64–89 (1968)

    MathSciNet  MATH  Google Scholar 

  61. Thaler, M.: Estimates of the invariant densities of endomorphisms with indifferent fixed points. Isr. J. Math. 37(4), 303–314 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  62. Toland, J.: Dual of \(L^\infty (X,\cal{L},\lambda )\) Finitely Additive Measures and Weak Convergence: A Primer. Springer, Berlin (2020)

    Book  MATH  Google Scholar 

  63. van Enter, A.C.D., Fernández, R., Sokal, A.D.: Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory. J. Stat. Phys. 72(5–6), 879–1167 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  64. Walters, P.: A variational principle for the pressure of continuous transformations. Am. J. Math. 97(4), 937–971 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  65. Walters, P.: Invariant measures and equilibrium states for some mappings which expand distances. Trans. Am. Math. Soc. 236, 121–153 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  66. Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics, vol. 79. Springer, New York (1982)

  67. Walters, P.: Convergence of the Ruelle operator for a function satisfying Bowen’s condition. Trans. Am. Math. Soc. 353(1), 327–347 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  68. Walters, P.: A natural space of functions for the Ruelle operator theorem. Ergod. Theory Dyn. Syst. 27(4), 1323–1348 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  69. Yosida, K., Hewitt, E.: Finitely additive measures. Trans. Am. Math. Soc. 72, 46–66 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  70. Yuri, M.: On the convergence to equilibrium states for certain non-hyperbolic systems. Ergod. Theory Dyn. Syst. 17(4), 977–1000 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

L. Cioletti and R. Ruviaro were financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES)—Finance Code 001. L. Cioletti and R. Ruviaro acknowledges the CNPq through projects PQ 313217/2018-1, PQ 316386/2021-9 DPP and FAP-DF for the financial support. We thank the anonymous referees for their helpful comments, and constructive remarks, and also thank Evgeny Verbitskiy for pointing out an error in a previous version of Theorem 5 and for bringing to our attention the example in Sect. 5.

Author information

Authors and Affiliations

  1. Universidade de Brasília, MAT, Brasília, 70910-900, Brazil

    L. Cioletti & R. Ruviaro

  2. Johann Bernoulli Instituut, Rijksuniversiteit Groningen, Nijenborgh 9, 9747 AG, Groningen, The Netherlands

    A. van Enter

Authors
  1. L. Cioletti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. van Enter
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Ruviaro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Cioletti.

Additional information

Communicated by H. Bruin.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cioletti, L., Enter, A.v. & Ruviaro, R. The double transpose of the Ruelle operator. Monatsh Math 200, 523–544 (2023). https://doi.org/10.1007/s00605-022-01818-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-022-01818-7

Keywords

Mathematics Subject Classification

Search

Navigation