Existence of Gibbs States and Maximizing Measures on a General One-Dimensional Lattice System with Markovian Structure

• Autores: Rafael Rigão Souza, Victor Vargas
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 1, 2022
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • Consider a compact metric space (M,dM) and X=MN. We prove a Ruelle’s Perron Frobenius Theorem for a class of compact subshifts with Markovian structure introduced in da Silva et al. (Bull Braz Math Soc 45:53–72, 2014) which are defined from a continuous function A:M×M→R that determines the set of admissible sequences. In particular, this class of subshifts includes the finite Markov shifts and models where the alphabet is given by the unit circle S1. Using the involution Kernel, we characterize the normalized eigenfunction of the Ruelle operator associated to its maximal eigenvalue and present an extension of its corresponding Gibbs state to the bilateral approach. From these results, we prove existence of equilibrium states and accumulation points at zero temperature in a particular class of countable Markov shifts.

• Referencias bibliográficas
  • 1. Aaronson, J.: An introduction to infinite ergodic theory, volume 50 of Mathematical Surveys and Monographs. American Mathematical Society,...
  • 2. Baraviera, A., Leplaideur, R., Lopes, A.: Ergodic optimization, zero temperature limits and the max-plus algebra. Publicações Matemáticas...
  • 3. Baraviera, A., Lopes, A.O., Thieullen, P.: A large deviation principle for the equilibrium states of Hölder potentials: the zero temperature...
  • 4. Baraviera, A.T., Cioletti, L.M., Lopes, A.O., Mohr, J., Souza, R.R.: On the general one-dimensional XY model: positive and zero temperature,...
  • 5. Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, vol. 470. Springer, Cham (1975)
  • 6. Brémont, J.: Gibbs measures at temperature zero. Nonlinearity 16(2), 419–426 (2003). https://doi.org/ 10.1088/0951-7715/16/2/303
  • 7. Chauta, J., Freire, R.: Peierls barrier for countable markov shifts. (2019). arXiv:1904.09655
  • 8. Chazottes, J.-R., Gambaudo, J.-M., Ugalde, E.: Zero-temperature limit of one-dimensional Gibbs states via renormalization: the case of...
  • 9. Cioletti, L., Denker, M., Lopes, A.O., Stadlbauer, M.: Spectral properties of the Ruelle operator for product-type potentials on shift...
  • 10. Cioletti, L.: Correlation inequalities and monotonicity properties of the ruelle operator. Stoch. Dyn 19(6), 1950048 (2019). https://doi.org/10.1142/S0219493719500485
  • 11. Cioletti, L., Silva, E.A., Stadlbauer, M.: Thermodynamic formalism for topological markov chains on standard borel spaces. Discrete Contin....
  • 12. Contreras, G., Lopes, A.O., Oliveira, E.R.: Ergodic transport theory, periodic maximizing probabilities and the twist condition. In Modeling,...
  • 13. da Silva, E.A., da Silva, R.R., Souza, R.R.A.: The analyticity of a generalized Ruelle‘s operator. Bull. Braz. Math. Soc. 45(1), 53–72...
  • 14. Freire, R., Vargas, V.: Equilibrium states and zero temperature limit on topologically transitive countable Markov shifts. Trans. Am....
  • 15. Iommi, G.: Ergodic optimization for renewal type shifts. Monatsh. Math. 150(2), 91–95 (2007). https:// doi.org/10.1007/s00605-005-0389-x
  • 16. Jenkinson, O., Mauldin, R.D., Urba ´nski, M.: Zero temperature limits of Gibbs-equilibrium states for countable alphabet subshifts of...
  • 17. Kempton, T.: Zero temperature limits of Gibbs equilibrium states for countable Markov shifts. J. Stat. Phys. 143(4), 795–806 (2011). https://doi.org/10.1007/s10955-011-0195-x
  • 18. Kitchens, B.P.: Symbolic dynamics. Universitext. Springer-Verlag, Berlin, (1998). One-sided, twosided and countable state Markov shifts
  • 19. Leplaideur, R.: A dynamical proof for the convergence of Gibbs measures at temperature zero. Nonlinearity 18(6), 2847–2880 (2005). https://doi.org/10.1088/0951-7715/18/6/023
  • 20. Leplaideur, R., Watbled, F.: Curie-weiss type models for general spin spaces and quadratic pressure in ergodic theory. J. Stat. Phys....
  • 21. Leplaideur, R., Watbled, F.: Generalized curie-weiss-potts model and quadratic pressure in ergodic theory. (2020). arXiv:2003.09535
  • 22. Lopes, A.O., Mengue, J.K., Mohr, J., Souza, R.R.: Entropy and variational principle for onedimensional lattice systems with a general...
  • 23. Lopes, A.O., Messaoudi, A., Stadlbauer, M., Vargas, V.: Invariant probabilities for discrete time linear dynamics via thermodynamic formalism....
  • 24. Lopes, A.O., Mohr, J., Souza, R.R., Thieullen, P.: Negative entropy, zero temperature and Markov chains on the interval. Bull. Braz. Math....
  • 25. Lopes, A.O., Vargas, V.: Gibbs states and gibbsian specifications on the space RN. Dyn. Syst. Int. J. 35(2), 216–241 (2020). https://doi.org/10.1080/14689367.2019.1663789
  • 26. Lopes, A.O., Vargas, V.: The ruelle operator for symmetric β-shifts. Publ. Math. 64(2), 661–680 (2020). https://doi.org/10.5565/PUBLMAT6422012
  • 27. Mauldin, R.D., Urba ´nski, M.: Gibbs states on the symbolic space over an infinite alphabet. Israel J. Math. 125, 93–130 (2001). https://doi.org/10.1007/BF02773377
  • 28. Mauldin, R.D., Urba ´nski, M.: Graph directed Markov systems, volume 148 of Cambridge Tracts in Mathematics. Cambridge University Press,...
  • 29. Mohr, J.: Product type potential on the xy model: selection of maximizing probability and a large deviation principle. (2019). arXiv:1805.09858
  • 30. Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 268, 187–188 (1990)
  • 31. Ruelle, D.: Statistical mechanics of a one-dimensional lattice gas. Commun. Math. Phys. 9, 267–278 (1968)
  • 32. Ruelle, D.: Thermodynamic formalism, volume 5 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Co., Reading,...
  • 33. Sarig, O.: Existence of Gibbs measures for countable Markov shifts. Proc. Am. Math. Soc. 131(6), 1751–1758 (2003). https://doi.org/10.1090/S0002-9939-03-06927-2
  • 34. Sarig, O.M.: Thermodynamic formalism for countable Markov shifts. Ergodic Theory Dyn. Syst. 19(6), 1565–1593 (1999). https://doi.org/10.1017/S0143385799146820
  • 35. Sina˘ı, Y.G.: Theory of phase transitions: rigorous results, volume 108 of International Series in Natural Philosophy. Pergamon Press,...
  • 36. Thompson, C.J.: Infinite-spin Ising model in one dimension. J. Math. Phys. 9, 241–245 (1968). https:// doi.org/10.1063/1.1664574
  • 37. van Enter, A.C.D., Ruszel, W.M.: Chaotic temperature dependence at zero temperature. J. Stat. Phys. 127(3), 567–573 (2007). https://doi.org/10.1007/s10955-006-9260-2