Journal of Statistical Physics

, Volume 57, Issue 1–2, pp 289–299 | Cite as

Hausdorff dimensions in two-dimensional maps and thermodynamic formalism

  • G. Paladin
  • S. Vaienti


We compute numerically the Hausdorff dimensions of the Gibbs measures on the invariant sets of Axiom A systems. In particular, we stress the existence of a measure which has maximal dimension and can be relevant for the ergodic properties of the system. For hyperbolic maps of the plane with constant Jacobianj, we apply the Bowen-Ruelle formula, using the relationF(β=dH−1)=lnj, which links the Hausdorff dimensiondH of an attractor to a free energy functionalF(β) defined in the thermodynamic formalism. We provide numerical evidence that this relation remains valid for some nonhyperbolic maps, such as the Hénon map.

Key words

Strange attractors thermodynamic formalism Gibbs states Hausdorff dimension generalized Lyapunov exponents 


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  1. 1.
    R. Bowen,Publ. Math. IHES 50:11 (1979); D. Ruelle,Prog. Phys. 7:351 (1983).Google Scholar
  2. 2.
    D. Ruelle,Thermodynamic Formalism (Addison-Wesley, Reading, Massachusetts, 1978).Google Scholar
  3. 3.
    H. Fujsaka,Prog. Theor. Phys. 70:1264 (1983); R. Benzi, G. Paladin, G. Parisi, and A. Vulpiani,J. Phys. A 18:2157 (1985).Google Scholar
  4. 4.
    J. L. Kaplan and J. A. Yorke, inLecture Notes in Mathematics, Vol. 730 (Springer, Berlin, 1978), p. 204.Google Scholar
  5. 5.
    G. Paladin and A. Vulpiani,Phys. Rep. 186:147 (1987), and references therein.Google Scholar
  6. 6.
    M. J. Feigenbaum,J. Stat. Phys. 46:919 (1987).Google Scholar
  7. 7.
    P. Collet, J. L. Lebowitz, and A. Porzio,J. Stat. Phys. 47:609 (1987).Google Scholar
  8. 8.
    D. Bessis, G. Paladin, G. Turchetti, and S. Vaienti,J. Stat. Phys. 51:109 (1988).Google Scholar
  9. 9.
    S. Vaienti,J. Stat. Phys. 56:403 (1989).Google Scholar
  10. 10.
    T. Bohr and T. Tel, Thermodynamics of fractals, inDirection of Chaos, Vol. 2 (World Scientific, Singapore, to appear).Google Scholar
  11. 11.
    G. Paladin, inUniversalities in Condensed Matter Physics, R. Jullien, L. Peliti, and R. Rammal, eds. (Springer-Verlag, 1988).Google Scholar
  12. 12.
    G. Paladin and S. Vaienti,J. Phys. A 21:4609 (1988).Google Scholar
  13. 13.
    J. P. Eckmann and D. Ruelle,Rev. Mod. Phys. 57:617 (1985).Google Scholar
  14. 14.
    D. Auerbach, P. Cvitanović, J. P. Eckmann, G. H. Gunaratne, and I. Procaccia,Phys. Rev. Lett. 58:2387 (1987); G. H. Gunaratne and I. Procaccia,Phys. Rev. Lett. 59:1377 (1987).Google Scholar
  15. 15.
    P. Cvitanović, Invariant measurements of strange set in terms of cycles. Preprint (1988).Google Scholar
  16. 16.
    L. Young,Ergodic Theory Dynam. Syst. 2:109 (1982).Google Scholar
  17. 17.
    Ya. B. Pesin,Ergodic Theory Dynam. Syst. 5:595 (1985).Google Scholar
  18. 18.
    L. P. Kadanoff and C. Tang,Proc. Natl. Acad. Sci. USA 81:1276 (1984); H. Kantz and P. Grassberger,Physica D 17:75 (1985).Google Scholar
  19. 19.
    P. Collet and Y. Levry,Commun. Math. Phys. 93:461 (1984).Google Scholar
  20. 20.
    P. Grassberger, A. Politi, and R. Badii,J. Stat. Phys. 51:135 (1988).Google Scholar
  21. 21.
    A. Politi, R. Badii, and P. Grassberger,J. Phys. A 21:L763 (1988).Google Scholar
  22. 22.
    P. Walters,Am. J. Math. 97:937 (1975).Google Scholar
  23. 23.
    H. McCluskey and A. Manning,Ergodic Theory Dynam. Syst. 3:251 (1983).Google Scholar

Copyright information

© Plenum Publishing Corporation 1989

Authors and Affiliations

  • G. Paladin
    • 1
    • 2
  • S. Vaienti
    • 3
    • 4
  1. 1.Dipartimento di FisicaUniversità “La Sapienza”RomeItaly
  2. 2.GNSM-CISM Unità di RomaRomeItaly
  3. 3.Centre Physique Théorique (Laboratoire propre du CNRS)Marseille 09France
  4. 4.Dipartimento di FisicaUniversità di BolognaBolognaItaly

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