Equilibrium State for One-Dimensional Lorenz-Like Expanding Maps

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Abstract

Let \(L:[0,1]{\setminus }\{d\}\rightarrow [0,1]\) be a one-dimensional Lorenz-like expanding map (d is the point of discontinuity), \(\mathcal {P}=\{ (0,d),(d,1) \}\) and \(C^{\alpha }([0,1],{\mathcal {P}})\) the set of piecewise Hölder-continuous potentials of [0, 1] with the usual \(\mathcal {C}^0\) topology. In this context, applying a criteria by Buzzi and Sarig (Ergod Theory Dyn Syst 23(5):1383–1400, 2003, Th. 1.3), we prove that there exists an open and dense subset \(\mathcal {H}\) of \(C^{\alpha }([0,1],{\mathcal {P}})\), such that each \(\phi \in \mathcal {H}\) admits exactly one equilibrium state.

Keywords

Equilibrium state Lorenz Maps 

Mathematics Subject Classification

Primary 37D25 Secondary 37D30 37D20 

Notes

Acknowledgements

We thank Ali Tahzibi for proposing the problem and for many helpful suggestions during the preparation of the paper. Also, we thank Daniel Smania and Krerley Oliveira for some helpful conversations and comments on the problem.

References

  1. Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics, vol. 470. Springer, Berlin (1975)Google Scholar
  2. Bruin, H.: For almost every tent map, the turning point is typical. Fund. Math. 155(3), 215–235 (1998)MathSciNetMATHGoogle Scholar
  3. Buzzi, Jérôme: Entropy of equilibrium measures of continuous piecewise monotonic maps. Stoch. Dyn. 4(1), 84–94 (2004)MathSciNetCrossRefMATHGoogle Scholar
  4. Buzzi, J., Sarig, O.: Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps. Ergod. Theory Dyn. Syst. 23(5), 1383–1400 (2003)MathSciNetCrossRefMATHGoogle Scholar
  5. Chazottes, Jean-René., Keller, G.: Pressure and equilibrium states in ergodic theory. In: Mathematics of Complexity and Dynamical Systems. 1–3, pp. 1422–1437. Springer, New York (2012)Google Scholar
  6. Climenhaga, V., Thompson, D.J.: Equilibrium states beyond specification and the Bowen property. J. Lond. Math. Soc. (2) 87(2), 401–427 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. Climenhaga, V., Thompson, D.J., Yamamoto, K.: Large deviations for systems with non-uniform structure. Trans. Am. Math. Soc. 369(6), 4167–4192 (2017)MathSciNetCrossRefMATHGoogle Scholar
  8. Denker, M., Urbański, M.: Ergodic theory of equilibrium states for rational maps. Nonlinearity 4(1), 103–134 (1991)MathSciNetCrossRefMATHGoogle Scholar
  9. Denker, M., Keller, G., Urbański, M.: On the uniqueness of equilibrium states for piecewise monotone mappings. Studia Math. 97(1), 27–36 (1990)MathSciNetCrossRefMATHGoogle Scholar
  10. Denker, M., Przytycki, F., Urbański, M.: On the transfer operator for rational functions on the Riemann sphere. Ergod. Theory Dyn. Syst. 16(2), 255–266 (1996)MathSciNetCrossRefMATHGoogle Scholar
  11. Faller, B., Pfister, C.-E.: A point is normal for almost all maps \(\beta x+\alpha \) mod 1 or generalized \(\beta \)-transformations. Ergod. Theory Dyn. Syst. 29(5), 1529–1547 (2009)CrossRefMATHGoogle Scholar
  12. Glendinning, P.: Topological conjugation of Lorenz maps by \(\beta \)-transformations. Math. Proc. Camb. Philos. Soc. 107(2), 401–413 (1990)MathSciNetCrossRefMATHGoogle Scholar
  13. Graczyk, J., Swipolhk A., G.: The real Fatou conjecture. Ann. Math. Stud. vol. 144. Princeton University Press, Princeton (1998)Google Scholar
  14. Guckenheimer, J.: A strange, strange attractor, in the hopf bifurcation theorem and its applications. In: Marsden, J., McCracken, M. (Eds.), pp. 368–381. Springer (1976)Google Scholar
  15. Guckenheimer, J., Williams, R.F.: Structural stability of Lorenz attractors. Inst. Hautes Études Sci. Publ. Math. 50, 59–72 (1979)MathSciNetCrossRefMATHGoogle Scholar
  16. Haydn, N.: Convergence of the transfer operator for rational maps. Ergod. Theory Dyn. Syst. 19(3), 657–669 (1999)MathSciNetCrossRefMATHGoogle Scholar
  17. Hofbauer, Franz.: Examples for the nonuniqueness of the equilibrium state. Trans. Am. Math. Soc. 228, 223–241 (1977)Google Scholar
  18. Hofbauer, F.: On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. Isr. J. Math. 34(3), 213–237 (1980) (1979)Google Scholar
  19. Hofbauer, F.: A function with countably many ergodic equilibrium states. Math. Z. 154(3), 275–281 (1977)MathSciNetCrossRefMATHGoogle Scholar
  20. Hofbauer, F.: The maximal measure for linear mod one transformations. J. Lond. Math. Soc. (2) 23(1), 92–112 (1981)MathSciNetCrossRefMATHGoogle Scholar
  21. Hofbauer, F., Keller, G.: Equilibrium states for piecewise monotonic transformations. Ergod. Theory Dyn. Syst. 2(1), 23–43 (1982)MathSciNetCrossRefMATHGoogle Scholar
  22. Hofbauer, F., Keller, G.: Equilibrium states and Hausdorff measures for interval maps. Math. Nachr. 164, 239–257 (1993)MathSciNetCrossRefMATHGoogle Scholar
  23. Inoquio-Renteria, I., Rivera-Letelier, J.: A characterization of hyperbolic potentials of rational maps. Bull. Braz. Math. Soc. (N.S.) 43(1), 99–127 (2012)MathSciNetCrossRefMATHGoogle Scholar
  24. Keller, G.: Equilibrium States in Ergodic Theory. London Mathematical Society Student Texts, vol. 42. Cambridge University Press, Cambridge (1998)Google Scholar
  25. Keller, G.: Generalized bounded variation and applications to piecewise monotonic transformations. Z. Wahrsch. Verw. Gebiete 69(3), 461–478 (1985)MathSciNetCrossRefMATHGoogle Scholar
  26. Li, H., Rivera-Letelier, J.: Equilibrium states of interval maps for hyperbolic potentials. Nonlinearity 27(8), 1779–1804 (2014)MathSciNetCrossRefMATHGoogle Scholar
  27. Li, H., Rivera-Letelier, J.: Equilibrium states of weakly hyperbolic one-dimensional maps for Hölder potentials. Commun. Math. Phys. 328(1), 397–419 (2014)CrossRefMATHGoogle Scholar
  28. Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci 20, 130–141 (1963)CrossRefMATHGoogle Scholar
  29. Palmer, M. R.: On classification of Measure Preserving Transformations of Lebesgue Spaces. Ph.D. thesis, University of Warwick, (1979)Google Scholar
  30. Parry, W.: The Lorenz attractor and a related population model. In: Ergodic theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978), volume 729 of Lecture Notes in Math., pp. 169–187. Springer, Berlin (1979)Google Scholar
  31. Parry, W.: Symbolic dynamics and transformations of the unit interval. Trans. Am. Math. Soc. 122, 368–378 (1966)MathSciNetCrossRefMATHGoogle Scholar
  32. Przytycki, F.: On the Perron-Frobenius-Ruelle operator for rational maps on the Riemann sphere and for Hölder continuous functions. Bol. Soc. Brasil. Mat. (N.S.) 20(2), 95–125 (1990)MathSciNetCrossRefMATHGoogle Scholar
  33. Rand, D.: The topological classifications of the lorenz attractor. Math. Proc. Camb. Philos. Soc. 83, 451–460 (1978)MathSciNetCrossRefMATHGoogle Scholar
  34. Robinson, R.C.: An introduction to dynamical systems—continuous and discrete, volume 19 of Pure and Applied Undergraduate Texts, 2nd edn. American Mathematical Society, Providence, RI (2012)Google Scholar
  35. Ruelle, D.: Thermodynamic formalism, volume 5 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Co., Reading, Mass., The mathematical structures of classical equilibrium statistical mechanics, With a foreword by Giovanni Gallavotti and Gian-Carlo Rota (1978)Google Scholar
  36. Sparrow, C.: The Lorenz equations: bifurcations, chaos, and strange attractors. Applied Mathematical Sciences, vol. 41. Springer, New York (1982)Google Scholar
  37. Walters, P.: An introduction to Ergodic Theory. Graduate Texts in Mathematics, vol. 79. Springer, New York (1982)Google Scholar
  38. Williams, R.F.: The structure of Lorenz attractors. Inst. Hautes Études Sci. Publ. Math. 50, 73–99 (1979)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2018

Authors and Affiliations

  1. 1.Faculdade de MatemáticaFAMAT-UFUUberlândiaBrazil

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