Communications in Mathematical Physics

, Volume 300, Issue 1, pp 65–94 | Cite as

Natural Equilibrium States for Multimodal Maps

Article

Abstract

This paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps. This class contains, but it is larger than, Collet-Eckmann. For a map in this class, we prove existence and uniqueness of equilibrium states for the geometric potentials −t log |Df|, for the largest possible interval of parameters t. We also study the regularity and convexity properties of the pressure function, completely characterising the first order phase transitions. Results concerning the existence of absolutely continuous invariant measures with respect to the Lebesgue measure are also obtained.

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References

  1. BC.
    Benedicks M., Carleson L.: On iterations of 1 − ax 2 on (−1, 1). Ann. of Math. 122, 1–25 (1985)CrossRefMathSciNetGoogle Scholar
  2. Bi.
    Billingsley P.: Probability and measure. Second edition. John Wiley and Sons, New York (1986)MATHGoogle Scholar
  3. Bo.
    Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Springer Lect. Notes in Math. 470, Berlin-Heidelberg-New York: Springer, 1975Google Scholar
  4. BB.
    Brucks, K.M., Bruin, H.: Topics from one-dimensional dynamics. London Mathematical Society Student Texts 62. Cambridge: Cambridge University Press, 2004Google Scholar
  5. B1.
    Bruin H.: Induced maps, Markov extensions and invariant measures in one–dimensional dynamics. Commun. Math. Phys. 168, 571–580 (1995)MATHCrossRefMathSciNetADSGoogle Scholar
  6. B2.
    Bruin H.: Minimal Cantor systems and unimodal maps. J. Difference Eq. Appl. 9, 305–318 (2003)MATHCrossRefMathSciNetADSGoogle Scholar
  7. BK.
    Bruin H., Keller G.: Equilibrium states for S-unimodal maps. Erg. Th. Dynam. Syst. 18, 765–789 (1998)MATHCrossRefMathSciNetGoogle Scholar
  8. BKNS.
    Bruin H., Keller G., Nowicki T., van Strien S.: Wild Cantor attractors exist. Ann. of Math. 143(2), 97–130 (1996)MATHCrossRefMathSciNetGoogle Scholar
  9. BS.
    Bruin H., van Strien S.: Expansion of derivatives in one–dimensional dynamics. Israel. J. Math. 137, 223–263 (2003)MATHCrossRefMathSciNetGoogle Scholar
  10. BT1.
    Bruin H., Todd M.: Equilibrium states for potentials with \({\sup \varphi-\inf \varphi < h_{top}(f)}\) . Commun. Math. Phys. 283, 579–611 (2008)MATHCrossRefMathSciNetADSGoogle Scholar
  11. BT2.
    Bruin H., Todd M.: Equilibrium states for interval maps: the potential −tlog|Df|. Ann. Sci. École Norm. Sup. 42(4), 559–600 (2009)MATHMathSciNetGoogle Scholar
  12. BT3.
    Bruin H., Todd M.: Return time statistics for invariant measures for interval maps with positive Lyapunov exponent. Stoch. Dyn. 9, 81–100 (2009)MATHCrossRefMathSciNetGoogle Scholar
  13. BuS.
    Buzzi J., Sarig O.: Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps. Erg. Th. Dynam. Systs. 23, 1383–1400 (2003)MATHCrossRefMathSciNetGoogle Scholar
  14. C.
    Cederval, S.: Invariant measures and correlation decay for S-multimodal interval maps. PhD thesis, Imperial College, 2006Google Scholar
  15. CRL.
    Cortez M.I., Rivera-Letelier J.: Invariant measures of minimal post-critical sets of logistic maps. Israel J. Math. 176, 157–193 (2010)MATHCrossRefGoogle Scholar
  16. DU.
    Denker M., Urbanski M.: Ergodic theory of equilibrium states for rational maps. Nonlinearity 4, 103–134 (1991)MATHCrossRefMathSciNetADSGoogle Scholar
  17. D1.
    Dobbs, N.: Critical points, cusps and induced expansion in dimension one. Thesis, Université Paris-Sud, Orsay, 2006Google Scholar
  18. D2.
    Dobbs N.: Visible measures of maximal entropy in dimension one. Bull. Lond. Math. Soc. 39, 366–376 (2007)MATHCrossRefMathSciNetGoogle Scholar
  19. D3.
    Dobbs N.: Renormalisation induced phase transitions for unimodal maps. Commun. Math. Phys. 286, 377–387 (2009)MATHCrossRefMathSciNetADSGoogle Scholar
  20. D4.
    Dobbs, N.: On cusps and flat tops. http://arXiv.org/abs/0801.3815v1[math.DS], 2008
  21. Dobr.
    Dobrušhin R.L.: Description of a random field by means of conditional probabilities and conditions for its regularity. Teor. Verojatnost. i Primenen 13, 201–229 (1968)MathSciNetGoogle Scholar
  22. E.
    Ellis, R.S.: Entropy, large deviations, and statistical mechanics. Classics in Mathematics, Berlin: Springer-Verlag, 2006Google Scholar
  23. FT.
    Frietas J., Todd M.: Statistical stability of equilibrium states for interval maps. Nonlinearity 22, 259–281 (2009)CrossRefMathSciNetADSGoogle Scholar
  24. GS.
    Graczyk J., Swia̧tek G.: Generic hyperbolicity in the logistic family. Ann. Math. 146, 1–52 (1997)MATHCrossRefMathSciNetGoogle Scholar
  25. Gu1.
    Gurevič B.M.: Topological entropy for denumerable Markov chains. Dokl. Akad. Nauk SSSR 10, 911–915 (1969)Google Scholar
  26. Gu2.
    Gurevič B.M.: Shift entropy and Markov measures in the path space of a denumerable graph. Dokl. Akad. Nauk SSSR 11, 744–747 (1970)Google Scholar
  27. H.
    Hofbauer F.: Piecewise invertible dynamical systems. Probab. Theory Relat. Fields 72, 359–386 (1986)MATHCrossRefMathSciNetGoogle Scholar
  28. J.
    Jakobson M.V.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81, 39–88 (1981)MATHCrossRefMathSciNetADSGoogle Scholar
  29. Je.
    Jenkinson O.: Ergodic optimization. Discrete Contin. Dyn. Syst. 15, 197–224 (2006)MATHCrossRefMathSciNetGoogle Scholar
  30. KH.
    Katok, A., Hasselblat, B.: Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications, 54. Cambridge: Cambridge University Press, 1995Google Scholar
  31. K1.
    Keller G.: Lifting measures to Markov extensions. Monatsh. Math. 108, 183–200 (1989)MATHCrossRefMathSciNetGoogle Scholar
  32. K2.
    Keller, G.: Equilibrium states in ergodic theory. London Mathematical Society Student Texts, 42. Cambridge: Cambridge University Press, 1998Google Scholar
  33. KStP.
    Keller, G., Pierre, M. St.: Topological and measurable dynamics of Lorenz maps. In: Ergodic theory, analysis, and efficient simulation of dynamical systems, Berlin: Springer, 2001, pp. 333–361Google Scholar
  34. L.
    Ledrappier F.: Some properties of absolutely continuous invariant measures on an interval. Erg. Th. Dynam. Syst. 1, 77–93 (1981)MATHMathSciNetGoogle Scholar
  35. Ly1.
    Lyubich M.: Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. of Math. 140(2), 347–404 (1994)MATHCrossRefMathSciNetGoogle Scholar
  36. Ly2.
    Lyubich M.: Dynamics of quadratic polynomials I-II. Acta Math. 178, 185–297 (1997)MATHCrossRefMathSciNetGoogle Scholar
  37. MS.
    Makarov N., Smirnov S.: On thermodynamics of rational maps. II. Non-recurrent maps. J. London Math. Soc. 67(2), 417–432 (2003)MATHCrossRefMathSciNetGoogle Scholar
  38. MU1.
    Mauldin R., Urbański M.: Dimensions and measures in infinite iterated function systems. Proc. London Math. Soc. 73(3), 105–154 (1996)MATHCrossRefMathSciNetGoogle Scholar
  39. MU2.
    Mauldin R., Urbański M.: Gibbs states on the symbolic space over an infinite alphabet. Israel J. Math. 125, 93–130 (2001)MATHCrossRefMathSciNetGoogle Scholar
  40. MN.
    Melbourne I., Nicol M.: Large deviations for nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 360, 6661–6676 (2008)MATHCrossRefMathSciNetGoogle Scholar
  41. MvS.
    de Melo, W., van Strien, S.: One dimensional dynamics. Ergebnisse Series 25, Berlin-Heidelberg-New York: Springer–Verlag, 1993Google Scholar
  42. NS.
    Nowicki T., Sands D.: Non-uniform hyperbolicity and universal bounds for S-unimodal maps. Invent. Math. 132, 633–680 (1998)MATHCrossRefMathSciNetADSGoogle Scholar
  43. P.
    Pesin, Y.: Dimension Theory in Dynamical Systems. Cambridge: Cambridge Univ. Press, 1997Google Scholar
  44. PS.
    Pesin Y., Senti S.: Equilibrium measures for maps with inducing schemes. J. Mod. Dyn. 2, 1–31 (2008)MathSciNetGoogle Scholar
  45. MP.
    Pomeau Y., Manneville P.: Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74, 189–197 (1980)CrossRefMathSciNetADSGoogle Scholar
  46. PreS1.
    Prellberg T., Slawny J.: Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions. J. Stat. Phys. 66, 503–514 (1992)MATHCrossRefMathSciNetADSGoogle Scholar
  47. Pr.
    Przytycki F.: Lyapunov characteristic exponents are nonnegative. Proc. Amer. Math. Soc. 119, 309–317 (1993)MATHMathSciNetGoogle Scholar
  48. PrR.
    Przytycki, F., Rivera-Letelier, J.: Nice inducing schemes and the thermodynamics of rational maps. Preprint, http://arXiv.org/abs/0806.4385v2[math.DS], 2008
  49. RY.
    Rey-Bellet L., Young L.-S.: Large deviations in non-uniformly hyperbolic dynamical systems. Erg. Th. Dynam. Syst. 28, 587–612 (2008)MATHMathSciNetGoogle Scholar
  50. Roc.
    Rockafellar, R.T.: Convex analysis. Princeton Mathematical Series, No. 28, Princeton, N.J.: Princeton University Press, 1970Google Scholar
  51. Rov.
    Rovella A.: The dynamics of perturbations of the contracting Lorenz attractor. Bol. Soc. Brasil. Mat. (N.S.) 24, 233–259 (1993)MATHCrossRefMathSciNetGoogle Scholar
  52. Roy.
    Royden H.L.: Real analysis Third edition. Macmillan Publishing Company, New York (1988)MATHGoogle Scholar
  53. Ru1.
    Ruelle D.: An inequality for the entropy of differentiable maps. Bol. Soc. Brasil. Mat. 9, 83–87 (1978)MATHCrossRefMathSciNetGoogle Scholar
  54. Ru2.
    Ruelle, D.: Thermodynamic formalism. The mathematical structures of classical equilibrium statistical mechanics. With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. Encyclopedia of Mathematics and its Applications, 5. Reading, MA: Addison-Wesley Publishing Co., 1978Google Scholar
  55. S1.
    Sarig O.: Thermodynamic formalism for countable Markov shifts. Erg. Th. Dynam. Syst. 19, 1565–1593 (1999)MATHCrossRefMathSciNetGoogle Scholar
  56. S2.
    Sarig O.: On an example with topological pressure which is not analytic. C.R. Acad. Sci. Serie I: Math. 330, 311–315 (2000)MATHMathSciNetADSGoogle Scholar
  57. S3.
    Sarig O.: Phase transitions for countable Markov shifts. Commun. Math. Phys. 217, 555–577 (2001)MATHCrossRefMathSciNetADSGoogle Scholar
  58. S4.
    Sarig O.: Existence of Gibbs measures for countable Markov shifts. Proc. Amer. Math. Soc. 131, 1751–1758 (2003)MATHCrossRefMathSciNetGoogle Scholar
  59. S5.
    Sarig O.: Critical exponents for dynamical systems. Commun. Math. Phys. 267, 631–667 (2006)MATHCrossRefMathSciNetADSGoogle Scholar
  60. Si.
    Sinai J.G.: Gibbs measures in ergodic theory. Uspehi Mat. Nauk 27, 21–64 (1972)MATHMathSciNetGoogle Scholar
  61. T.
    Todd, M.: Multifractal analysis for multimodal maps. Preprint http://arXiv.org/abs/0809.1074v3[math-DS], 2009
  62. Wa.
    Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics 79, Berlin-Heidelberg-New York: Springer, 1981Google Scholar
  63. Y.
    Young L.S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Facultad de MatemáticasPontificia Universidad Católica de Chile (PUC)SantiagoChile
  2. 2.Departamento de Matemática PuraFaculdade de Ciências da Universidade do PortoPortoPortugal
  3. 3.Department of Mathematics and StatisticsBoston UniversityBostonUSA

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