Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 1451–1494 | Cite as

Pesin’s entropy formula for \(C^1\) non-uniformly expanding maps

  • Vítor AraujoEmail author
  • Felipe Santos


We prove existence of equilibrium states with special properties for a class of distance expanding local homeomorphisms on compact metric spaces and continuous potentials. Moreover, we formulate a C\(^1\) generalization of Pesin’s Entropy Formula: all ergodic weak-SRB-like measures satisfy Pesin’s Entropy Formula for \(C^1\) non-uniformly expanding maps. We show that for weak-expanding maps such that \({\text {Leb}}\)-a.e x has positive frequency of hyperbolic times, then all the necessarily existing ergodic weak-SRB-like measures satisfy Pesin’s Entropy Formula and are equilibrium states for the potential \(\psi =-\log |\det Df|\). In particular, this holds for any \(C^1\)-expanding map and, in this case, the set of invariant probability measures that satisfy Pesin’s Entropy Formula is the weak\(^*\)-closed convex hull of the ergodic weak-SRB-like measures.


Non-uniform expansion SRB/physical-like measures Equilibrium states Pesin’s entropy formula \(C^1\) smooth Uniform expansion 

Mathematics Subject Classification

Primary 37D25 Secondary 37D35 37D20 37C40 



This is the PhD thesis of F. Santos at the Instituto de Matematica e Estatistica-Universidade Federal da Bahia (UFBA) under a CAPES scholarship. He thanks the Mathematics and Statistics Institute at UFBA for the use of its facilities and the finantial support from CAPES during his M.Sc. and Ph.D. studies. The authors thank the anonymous referee for the careful reading and the many useful suggestions that greatly helped to improve the quality of the text.


  1. 1.
    Alves, J.: Statistical analysis of non-uniformly expanding dynamical systems. Publicações Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2003. XXIV Colóquio Brasileiro de Matemática. [24th Brazilian Mathematics Colloquium]Google Scholar
  2. 2.
    Alves, J.F.: SRB measures for non-hyperbolic systems with multidimensional expansion. Annales scientifiques de l’Ecole normale supérieure 33(1), 1–32 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Alves, J.F., Araújo, V.: Hyperbolic times: frequency versus integrability. Ergod. Theory Dyn. Syst. 24, 1–18 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Alves, J.F., Bonatti, C., Viana, M.: SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140(2), 351–398 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Alves, J. F., Ramos, V., Siqueira, J.: Equilibrium stability for non-uniformly hyperbolic maps. ArXiv e-prints, (July 2017)Google Scholar
  6. 6.
    Araújo, V., Pacifico, M.: Large deviations for non-uniformly expanding maps. J. Stat. Phys. 125(2), 411–453 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ashley, J., Kitchens, B., Stafford, M.: Boundaries of markov partitions. Trans. Am. Math. Soc. 333(1), 177–201 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Avila, A., Bochi, J.: Generic expanding maps without absolutely continuous invariant \(\sigma \)-finite measure. Nonlinearity 19, 2717–2725 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Barreira, L., Pesin, Y.: Introduction to Smooth Ergodic Theory, Volume 148 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2013)zbMATHCrossRefGoogle Scholar
  10. 10.
    Bessa, M., Varandas, P.: On the entropy of conservative flows. Qual. Theory Dyn. Syst. 11, 11–22 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Bowen, R.: A horseshoe with positive measure. Invent. Math. 29, 203–204 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms, volume 470 of Lect. Notes in Math. Springer (1975)Google Scholar
  13. 13.
    Bowen, R., Ruelle, D.: The ergodic theory of Axiom A flows. Invent. Math. 29, 181–202 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Cao, Y., Yang, D.: On pesin’s entropy formula for dominated splittings without mixed behavior. J. Differ. Equ. 261(7), 3964–3986 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Catsigeras, E.: On Ilyashenko’s statistical attractors. Dyn. Syst. 29(1), 78–97 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Catsigeras, E.: Empiric stochastic stability of physical and pseudo-physical measures. ArXiv e-prints, (July 2017)Google Scholar
  17. 17.
    Catsigeras, E., Cerminara, M., Enrich, H.: The Pesin Entropy Formula for diffeomorphisms with dominated splitting. Ergod. Theory Dyn. Syst. 35(03), 737–761 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Catsigeras, E., Cerminara, M., Enrich, H.: Weak pseudo-physical measures and Pesin’s Entropy Formula for Anosov \(C^1\) diffeomorphisms. Contemp. Math. 698, 69–89 (2017)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Catsigeras, E., Enrich, H.: SRB-like measures for \(C^0\) dynamics. Bull. Pol. Acad. Sci. Math. 59(2), 151–164 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Catsigeras, E., Enrich, H.: Equilibrium states and SRB-like measures of \(C^1\)-expanding maps of the circle. Port. Math. 69, 193–212 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Catsigeras, E., Troubetzkoy, S.: Pseudo-physical measures for typical continuous maps of the interval. ArXiv e-prints, (May 2017)Google Scholar
  22. 22.
    Coven, E. M., Reddy, W. L.: Positively expansive maps of compact manifolds. In: Nitecki, Z., Robinson, C. (eds.), Global Theory of Dynamical Systems: Proceedings of an International Conference Held at Northwestern University, Evanston, Illinois, June 18–22, 1979, pages 96–110. Springer, Heidelberg (1980)Google Scholar
  23. 23.
    Golenishcheva-Kutuzova, T., Kleptsyn, V.: Convergence of the Krylov–Bogolyubov procedure in Bowen’s example. Math. Notes 82(5), 608–618 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Hofbauer, F., Keller, G.: Quadratic maps without asymptotic measure. Commun. Math. Phys. 127, 319–337 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Keller, G.: Generalized bounded variation and applications to piecewise monotonic transformations. Z. Wahrsch. Verw. Gebiete 69(3), 461–478 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Keller, G.: Equilibrium States in Ergodic Theory, Volume 42 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  27. 27.
    Ledrappier, F., Young, L .S.: The metric entropy of diffeomorphisms I. Characterization of measures satisfying Pesin’s entropy formula. Ann. Math 122, 509–539 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Liu, P.D.: Pesin’s Entropy formula for endomorphisms. Nagoya Math. J. 150, 197–209 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Pesin, Y.B.: Characteristic Lyapunov exponents and smooth ergodic theory. Rus. Math. Surv. 324, 55–114 (1977)zbMATHCrossRefGoogle Scholar
  30. 30.
    Przytycki, F., Urbański, M.: Conformal fractals: ergodic theory methods, volume 371 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, (2010)Google Scholar
  31. 31.
    Qian, M., Zhu, S.: SRB measures and Pesin’s entropy formula for endomorphisms. Trans. Am. Math. Soc. 354(4), 1453–1471 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Qiu, H.: Existence and uniqueness of SRB measure on \(C^1\) generic hyperbolic attractors. Commun. Math. Phys. 302(2), 345–357 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Quasf, A.N.: Non-ergodicity for \(C^1\) expanding maps and \(g\)-measures. Ergod. Theory Dyn. Syst. 16(3), 531–543 (1996)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Ramos, V., Viana, M.: Equilibrium states for hyperbolic potentials. Nonlinearity 30(2), 825 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Ruelle, D.: A measure associated with Axiom A attractors. Am. J. Math. 98, 619–654 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Ruelle, D.: An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Mat. 9, 83–87 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Ruelle, D.: Thermodynamic formalism. Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition. The mathematical structures of equilibrium statistical mechanics (2004)Google Scholar
  38. 38.
    Sinai, Y.: Gibbs measures in ergodic theory. Russ. Math. Surv. 27, 21–69 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Sun, W., Tian, X.: Dominated splitting and Pesin’s Entropy Formula. Discrete Contin. Dyn. Syst. 32(4), 1421–1434 (2012)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Tahzibi, A.: \(C^1\)-generic Pesin’s Entropy Formula. Comptes Rendus Mathematique 335(12), 1057–1062 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Takahashi, Y.: Entropy functional (free energy) for dynamical systems and their random perturbations. N. Holl. Math. Libr. Stoch. Anal. 32, 437–467 (1984)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Takens, F.: Heteroclinic attractors: time averages and moduli of topological conjugacy. Bull. Braz. Math. Soc. 25, 107–120 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Thaler, M.: Transformations on \([0,1]\) with infinite invariant measures. Isr. J. Math. 46(1–2), 67–96 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Varandas, P., Viana, M.: Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2), 555–593 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Viana, M., Oliveira, K.: Foundations of ergodic theory, volume 151 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, (2016)Google Scholar
  46. 46.
    Walters, P.: An introduction to ergodic theory, volume 79 of Graduate Texts in Mathematics. Springer, New York (1982)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Instituto de MatemáticaUniversidade Federal da Bahia Av. Ademar de Barros s/nSalvadorBrazil
  2. 2.Centro de Formação de ProfessoresUniversidade Federal do Reconcavo da Bahia, Avenida NestorAmargosaBrazil

Personalised recommendations