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Communications in Mathematical Physics

, Volume 281, Issue 3, pp 711–751 | Cite as

Ergodic Theory of Parabolic Horseshoes

  • Mariusz Urbański
  • Christian WolfEmail author
Article

Abstract

In this paper we develop the ergodic theory for a horseshoe map f which is uniformly hyperbolic, except at one parabolic fixed point ω and possibly also on W s (ω). We call f a parabolic horseshoe map. In order to analyze dynamical and geometric properties of such horseshoes, by making use of induced maps, we establish, in the context of σ-finite measures, an appropriate version of the variational principle for continuous potentials with mild distortion defined on subshifts of finite type. Staying in this setting, we propose a concept of σ-finite equilibrium states (each classical probability equilibrium state is a σ-finite equilibrium state). We then study the unstable pressure function \({t \mapsto P(-t \log |Df| E^u|)}\), the corresponding finite and σ-finite equilibrium states and their associated conditional measures. The main idea is to relate the pressure function to the pressure of an embedded parabolic iterated function system and to apply the developed theory of the symbolic σ-finite thermodynamic formalism. We prove, in particular, an appropriate form of the Bowen-Ruelle-Manning-McCluskey formula, the existence of exactly two σ-finite ergodic conservative equilibrium states for the potential –t u log |Df|E u | (where t u denotes the unstable dimension), one of which is the Dirac δ-measure supported at the parabolic fixed point and the other being non-atomic. We also show that the conditional measures of this non-atomic equilibrium state on unstable manifolds, are equivalent to (finite and positive) packing measures, whereas the Hausdorff measures vanish. As an application of our results we obtain a classification for the existence of a generalized physical measure, as well as a criteria implying the non-existence of an ergodic measure of maximal dimension.

Keywords

Ergodic Theory Unstable Manifold Hausdorff Dimension Maximal Dimension Iterate Function System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. A.
    Aaronson, J.: An introduction to infinite ergodic theory. Mathematical Surveys and Monographs, 50. Providence, RI: Amer. Math. Soc., 1997Google Scholar
  2. BW1.
    Barreira L, Wolf C.: Measures of maximal dimension for hyperbolic diffeomorphisms. Commun. Math. Phys. 239, 93–113 (2003)zbMATHCrossRefADSMathSciNetGoogle Scholar
  3. BW2.
    Barreira L., Wolf C.: Pointwise dimension and ergodic decompositions. Erg. Theory Dyn. Syst. 26(3), 653–671 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  4. DU.
    Denker M., Urbański M.: On the existence of conformal measures. Trans. A.M.S. 328, 563–587 (1991)zbMATHCrossRefGoogle Scholar
  5. DV.
    Diaz L., Viana M.: Discontinuity of Hausdorff dimension and limit capacity on arcs of diffeomorphisms. Erg. Theory Dyn. Syst. 9(3), 403–425 (1989)zbMATHMathSciNetGoogle Scholar
  6. DGS.
    Denker, M., Grillenberger, C., Sigmund, K.: Ergodic theory on compact spaces, Lecture Notes in Math. 527. Berlin: Springer-Verlag, 1976Google Scholar
  7. F.
    Falconer K.: Fractal Geometry: Mathematical Foundations and applications. Wiley, New York (2003)zbMATHGoogle Scholar
  8. HMU.
    Hanus P., Mauldin D., Urbański M.: Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems. Acta Math. Hungarica 96, 27–98 (2002)zbMATHCrossRefGoogle Scholar
  9. J.
    Jenkinson O.: Rotation, Entropy, and Equilibrium States. Trans. Amer. Math. Soc. 353, 3713–3739 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  10. KFS.
    Kornfeld P., Fomin S.V., Sinai Y.G.: Ergodic Theory. Springer, Berlin-Heidelberg-New York (1982)Google Scholar
  11. K.
    Krengel U.: Entropy of conservative transformations. Z. Wahr. verw. Geb. 7, 161–181 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  12. Ma.
    Manning A.: A relation between Lyapunov exponents, Hausdorff dimension and entropy. Ergodic Theory Dy. Syst. 1(4), 451–459 (1982)MathSciNetGoogle Scholar
  13. MM.
    Manning A., McCluskey H.: Hausdorff dimension for Horseshoes. Erg. Theory Dyn. Syst. 3, 251–260 (1983)MathSciNetGoogle Scholar
  14. MU1.
    Mauldin D., Urbański M.: Parabolic iterated function systems. Erg. Th. & Dyna. Syst. 20, 1423–1447 (2000)zbMATHCrossRefGoogle Scholar
  15. MU2.
    Mauldin D., Urbański M.: Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets. Cambridge Univ. Press, Cambridge (2003)zbMATHGoogle Scholar
  16. Me.
    Mendoza L.: The entropy of C 2 surface diffeomorphisms in terms of Hausdorff dimension and a Lyapunov exponent. Erg. Theory Dyn. Syst. 5(2), 273–283 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  17. PU.
    Przytycki, F., Urbański, M.: Fractals in the Plane - Ergodic Theory Methods. To appear Cambridge Univ. Press, available on Urbański’s webpage, http://www.math.unt.edu/~urbanski/
  18. Ra.
    Rams M.: Measures of maximal dimension for linear horseshoes. Real Analysis Exchange 31, 55–62 (2006)zbMATHMathSciNetGoogle Scholar
  19. Ru.
    Ruelle D.: Thermodynamic formalism. Addison-Wesley, Reading, MA (1978)zbMATHGoogle Scholar
  20. U1.
    Urbański, M.: Parabolic Cantor sets. Preprint 1995, available on Urbański’s webpage, http://www.math.unt.edu/~urbanski/
  21. U2.
    Urbański M.: Parabolic Cantor sets. Fund. Math. 151, 241–277 (1996)MathSciNetzbMATHGoogle Scholar
  22. U3.
    Urbański M.: Geometry and ergodic theory of conformal nonrecurrent dynamics. Erg. Th. Dyn. Syst. 17, 1449–1476 (1997)CrossRefzbMATHGoogle Scholar
  23. U4.
    Urbański M.: Hausdorff measures versus equilibrium states of conformal infinite iterated function systems. Periodica Math. Hung. 37, 153–205 (1998)CrossRefzbMATHGoogle Scholar
  24. Wa.
    Walters, P.: An introduction to ergodic theory. Graduate Texts in Mathematics 79, Berlin-Heidelberg-New York: Springer, 1981Google Scholar
  25. Wo.
    Wolf C.: Generalized physical and SRB measures for hyperbolic diffeomorphisms. J. Stat. Phys. 122(6), 1111–1138 (2006)zbMATHCrossRefADSMathSciNetGoogle Scholar
  26. Y.
    Young L.-S.: Dimension, entropy and Lyapunov exponents. Erg. Theory Dyn. Syst. 2, 109–124 (1982)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of North TexasDentonUSA
  2. 2.Department of MathematicsWichita State UniversityWichitaUSA

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