Collectanea Mathematica

, Volume 70, Issue 2, pp 347–356 | Cite as

Tail entropy and hyperbolicity of measures

  • Gang LiaoEmail author


We study the relationship between the tail entropy and the hyperbolicity of invariant measures. An upper bound of the tail entropy is given in terms of the hyperbolic index.


Tail entropy Hyperbolicity Upper semi-continuity 

Mathematics Subject Classification

37A35 37D25 37C40 


  1. 1.
    Barreira, L., Pesin, Y.: Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents. Cambridge Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bowen, R.: Entropy expansive maps. Trans. Am. Math. Soc. 164, 323–331 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Burguet, D.: A direct proof of the tail variational principle and its extension to maps. Ergod. Theory Dyn. Syst. 29, 357–369 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Burguet, D., Liao, G., Yang, J.: Asymptotic $h$-expansiveness rate of $C^{\infty }$ maps. Proc. Lond. Math. Soc. 111, 381–419 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Buzzi, J.: Intrinsic ergodicity for smooth interval maps. Isr. J. Math. 100, 125–161 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Downarowicz, T.: Entropy structure. J. Anal. Math. 96, 57–116 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hirsch, M., Pugh, C., Shub, M.: Invariant Manifolds Volume 583 of Lect. Notes in Math. Springer, Berlin (1977)CrossRefGoogle Scholar
  8. 8.
    Liao, G., Viana, M., Yang, J.: The entropy conjecture for diffeomorphisms away from tangencies. J. Eur. Math. Soc. 6, 2043–2060 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Newhouse, S.: Continuity properties of entropy. Ann. Math. 129, 215–235 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Oseledets, V.I.: A multiplicative ergodic theorem. Trans. Mosc. Math. Soc. 19, 197–231 (1968)zbMATHGoogle Scholar
  11. 11.
    Pesin, Y.: Families of invariant manifolds corresponding to nonzero characteristic exponents. Math. USSR-Izv. 40, 1261–1305 (1976)CrossRefzbMATHGoogle Scholar
  12. 12.
    Pollicott, M.: Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds. Cambridge Univ. Press, Cambridge (1993)CrossRefzbMATHGoogle Scholar
  13. 13.
    Yomdin, Y.: Volume growth and entropy. Isr. J. Math. 57, 285–300 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Universitat de Barcelona 2018

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Center for Dynamical Systems and Differential EquationsSoochow UniversitySuzhouChina

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