Advertisement

Large Deviations for Equilibrium Measures and Selection of Subaction

  • Jairo K. Mengue
Article

Abstract

Given a Lipschitz function \(f:\{1,\ldots ,d\}^\mathbb {N} \rightarrow \mathbb {R}\), for each \(\beta >0\) we denote by \(\mu _\beta \) the equilibrium measure of \(\beta f\) and by \(h_\beta \) the main eigenfunction of the Ruelle Operator \(L_{\beta f}\). Assuming that \(\{\mu _{\beta }\}_{\beta >0}\) satisfy a large deviation principle, we prove the existence of the uniform limit \(V= \lim _{\beta \rightarrow +\infty }\frac{1}{\beta }\log (h_{\beta })\). Furthermore, the expression of the deviation function is determined by its values at the points of the union of the supports of maximizing measures. We study a class of potentials having two ergodic maximizing measures and prove that a L.D.P. is satisfied. The deviation function is explicitly exhibited and does not coincide with the one that appears in the paper by Baraviera-Lopes-Thieullen which considers the case of potentials having a unique maximizing measure.

Keywords

Equilibrium measure Maximizing measure Large deviation principle 

References

  1. Baraviera, A., Leplaideur, R., Lopes, A.: Ergodic optimization, zero temperature limits and the max-plus algebra. IMPA, Rio de Janeiro (2013)zbMATHGoogle Scholar
  2. Baraviera, A., Lopes, A., Mengue, J.: On the selection of subaction and measure for a subclass of potentials defined by P. Walters. Ergodic Theory Dyn. Syst. 33, 1338–1362 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Baraviera, A., Lopes, A., Thieullen, P.H.: A large deviation principle for equilibrium states of Hölder potencials: the zero temperature case. Stoch. Dyn. 6, 77–96 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bissacot, R., Garibaldi, E., Thieullen, P.H.: Zero-temperature phase diagram for double-well type potentials in the summable variation class. Ergodic Theory Dyn. Syst. 1–23 (2016a). doi: 10.1017/etds.2016.57
  5. Bissacot, R., Mengue, J., Perez, E.: A large deviation principle for gibbs states on Countable Markov Shifts at Zero Temperature (2016b) (preprint)Google Scholar
  6. Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. 2nd Edition. Edited by Jean-René Chazottes. Springer-Verlag, Berlin (2008)Google Scholar
  7. Brémont, J.: Gibbs measures at temperature zero. Nonlinearity 16(2), 419–426 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Chazottes, J.R., Hochman, M.: On the zero-temperature limit of Gibbs states. Comm. Math. Phys 297(1), 265–281 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Contreras, G., Lopes, A.O.: Lyapunov minimizing measures for expanding maps of the circle. Ergodic Theory Dyn. Syst. 21, 1379–1409 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Conze, J. P., Guivarc’h, Y.: Croissance des sommes ergodiques et principe variationnel, manuscript circa (1993)Google Scholar
  11. Coronel, D., Rivera-Letelier, J.: Sensitive dependence of Gibbs measures at low temperatures. J. Stat. Phys. 160, 1658–1683 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Dembo, A., Zeitouni, O.: Large deviation techniques and applications. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  13. Jenkinson, O.: Ergodic optimization. Discrete Contin. Dyn. Syst. Ser. A 15, 197–224 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Kempton, T.: Zero temperature limits of Gibbs equilibrium states for countable Markov shifts. J. Stat. Phys. 143, 795–806 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Leplaideur, R.: A dynamical proof for the convergence of Gibbs measures at temperature zero. Nonlinearity 18, 2847–2880 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Leplaideur, R.: Flatness is a criterion for selection of maximizing measures. J. Stat. Phys. 147(4), 728–757 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Lopes, A., Mohr, J., Souza, R.: Negative Entropy, pressure and zero temperature: a LDP for stationary Markov Chains on \([0,1]\). Bull. Braz. Math. Soc. 40(1), 1–52 (2009)MathSciNetCrossRefGoogle Scholar
  18. Lopes, A., Mengue, J.: Selection of measure and a large deviation principle for the general XY model. Dyn. Syst. 29(1), 24–39 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Mengue, J.: Zeta-medidas e princípio dos grandes desvios. PhD thesis, UFRGS (2010). http://hdl.handle.net/10183/26002
  20. Morris, I.: A sufficient condition for the subordination principle in ergodic optimization. Bull. Lond. Math. Soc. 39(2), 214–220 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Morris, I.: Mather sets for sequences of matrices and applications to the study of joint spectral radii. Proc. Lond. Math. Soc. 107, 121–150 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque Vol 187–188 (1990)Google Scholar
  23. Walters, P.: A natural space of functions for the Ruelle operator theorem. Ergodic Theory Dyn. Syst. 27, 1323–1348 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2017

Authors and Affiliations

  1. 1.Universidade Federal do Rio Grande do SulPorto AlegreBrazil

Personalised recommendations