Journal of Statistical Physics

, Volume 51, Issue 1–2, pp 135–178 | Cite as

Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors

  • P. Grassberger
  • R. Badii
  • A. Politi
Articles

Abstract

The analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems. Prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective Liapunov exponents. For hyperbolic attractors, the latter determines the metric entropies and provides one constraint on the partial dimensions. For nonhyperbolic attractors, there are important modifications. We discuss them for the examples of the logistic and Hénon map. We show, in particular, that the generalized dimensions have singularities with noncontinuous derivative, similar to first-order phase transitions in statistical mechanics.

Key words

Dynamical systems generalized dimensions and entropies Liapunov exponents scaling functions hyperbolicity phase transitions 

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References

  1. 1.
    G. Parisi, Appendix, in U. Frisch, Fully developed turbulence and intermittency, inProceedings of International School on Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, M. Ghil, ed. (North-Holland, 1984); U. Frisch,Phys. Scripta T9:137 (1985).Google Scholar
  2. 2.
    R. Benzi, G. Paladin, G. Parisi, and A. Vulpiani,J. Phys. A 17:3521 (1984).Google Scholar
  3. 3.
    T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. Shraiman,Phys. Rev. A 33:1141 (1986); M. H. Jensen, L. P. Kadanoff, A. Libchaber, I. Procaccia, and J. Stavans,Phys. Rev. Lett. 55:2798 (1985).Google Scholar
  4. 4.
    D. Ruelle,Thermodynamic Formalism (Addison-Wesley, Reading. Massachusetts, 1978); O. Lanford, Entropy and equilibrium states in classical and statistical mechanics, inStatistical Mechanics and Mathematical Problems, A. Lenard, ed. (Springer, 1976).Google Scholar
  5. 5.
    B. B. Mandelbrot,The Fractal Geometry of Nature (Freeman, San Francisco, 1982).Google Scholar
  6. 6.
    P. Grassberger,Phys. Lett. 97A:227 (1983).Google Scholar
  7. 7.
    H. G. Hentschel and I. Procaccia,Physica 8D:435 (1983).Google Scholar
  8. 8.
    P. Grassberger,Phys. Lett. 107A:101 (1985).Google Scholar
  9. 9.
    V. N. Shtern,Dokl. Akad. Nauk SSSR 270:582 (1983).Google Scholar
  10. 10.
    J.-P. Eckmann and D. Ruelle,Rev. Mod. Phys. 57:617 (1985).Google Scholar
  11. 11.
    J. D. Farmer, E. Ott, and J. A. Yorke,Physica 7D:153 (1983).Google Scholar
  12. 12.
    J.-P. Eckmann and I. Procaccia,Phys. Rev. A 34:659 (1986); G. Paladin, L. Peliti, and A. Vulpiani, University of Rome, Preprint (1986).Google Scholar
  13. 13.
    V. Jakobson,Commun. Math. Phys. 81:39 (1981); M. Misiurewicz,Publ. Math. IHES 53:17 (1981).Google Scholar
  14. 14.
    M. Hénon,Commun. Math. Phys. 50:69 (1976).Google Scholar
  15. 15.
    S. Newhouse, Lectures on dynamical systems, inDynamical Systems (Birkhauser, Boston, 1980); J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, New York, 1986).Google Scholar
  16. 16.
    E. Ott, W. Withers, and J. A. Yorke,J. Stat. Phys. 36:687 (1984).Google Scholar
  17. 17.
    O. Rössler,Phys. Lett. 57A:397 (1976); P. Holmes,Phil. Trans. R. Soc. A 292:419 (1979).Google Scholar
  18. 18.
    A. Renyi,Probability Theory (North-Holland, Amsterdam, 1970).Google Scholar
  19. 19.
    R. Badii and A. Politi,Phys. Rev. Lett. 52:1661 (1984); R. Badii and A. Politi,J. Stat. Phys. 40:725 (1985).Google Scholar
  20. 20.
    L. P. Kadanoff, private communication.Google Scholar
  21. 21.
    J. Balatoni and A. Renyi, inSelected Papers of A. Renyi, Vol. 1, p. 558 (Akademia, Budapest, 1976).Google Scholar
  22. 22.
    S. J. Chang and P. R. Fendlay,Phys. Rev. A 33:4092 (1986).Google Scholar
  23. 23.
    D. Rand, The singularity spectrum for hyperbolic cantor sets and attractors, University of Arizona, preprint (1986); P. Collet, J. Lebowitz, and A. Porzio, Dimension spectrum for some dynamical systems, to be published.Google Scholar
  24. 24.
    P. Fredrickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke,J. Diff. Eqs. 49:185 (1983).Google Scholar
  25. 25.
    P. Grassberger, inChaos in Astrophysics, J. Perdanget al., eds. (Reiedl, Dortrecht, 1985); P. Grassberger, inChaos, A. V. Holden, ed. (Manchester University Press, Manchester, 1986).Google Scholar
  26. 26.
    P. Grassberger and I. Procaccia,Physica 13D:34 (1984).Google Scholar
  27. 27.
    P. Billingsley,Ergodic Theory and Information (Wiley, New York, 1965).Google Scholar
  28. 28.
    F. Takens, Invariants related to dimension and entropy, inAtas do 13° Coloquio Brasileiro de Matematica (1984).Google Scholar
  29. 29.
    P. Grassberger and I. Procaccia,Physica 9D:189 (1983).Google Scholar
  30. 30.
    J. D. Farmer, Order within chaos, Thesis, University of California, Santa Cruz (1981).Google Scholar
  31. 31.
    E. N. Lorenz,Physica 13D:90 (1984);17D:279 (1985).Google Scholar
  32. 32.
    R. Badii and A. Politi,Phys. Rev. 35A:1288 (1987).Google Scholar
  33. 33.
    H. Fujisaka,Prog. Theor. Phys. 70:1264 (1983).Google Scholar
  34. 34.
    G. Györgyi and P. Szepfalusy,Z. Phys. B 55:179 (1984); P. C. Hemmer,J. Phys. A 17:L247 (1984); S. Grossmann and H. Horner,Z. Phys. B 60:79 (1985).Google Scholar
  35. 35.
    B. V. Chirikov and D. L. Shepelyansky,Physica 13D:395 (1984); P. Grassberger and H. Kantz,Phys. Lett. 113A:167 (1985).Google Scholar
  36. 36.
    P. Grassberger,Physica 14D:365 (1985).Google Scholar
  37. 37.
    F. Ledrappier and L. S. Young,Ann. Math. 122:509 (1985).Google Scholar
  38. 38.
    Ya. B. Pesin,Russ. Math. Surv. 32:55 (1977); D. Ruelle,Ann. N. Y. Acad. Sci. 136:229 (1981).Google Scholar
  39. 39.
    R. Badii and A. Politi,Phys. Scripta 35:243 (1987).Google Scholar
  40. 40.
    G. Julia,J. Math. Ser. 7 (Paris) 4:47 (1918); P. Fatou,Bull. Soc. Math. France 47:161 (1919); H. Brolin,Ark. Mat. 6:103 (1965).Google Scholar
  41. 41.
    D. Ruelle,Ergod. Theory Dyn. Syst. 2:109 (1982); A. Manning, University of Warwick preprint (1984).Google Scholar
  42. 42.
    S. Ulam and J. Von Neumann,Bull. Am. Math. Soc. 53:1120 (1947).Google Scholar
  43. 43.
    L. de Arcangelis, S. Redner, and A. Coniglio,Phys. Rev. B 31:4725 (1985); R. Rammal, C. Tannous, and A.-M. S. Tremblay,Phys. Rev. A 31:2662 (1985).Google Scholar
  44. 44.
    S. Roux and C. D. Mitescu,Phys. Rev. B 35:898 (1987).Google Scholar
  45. 45.
    P. Cvitanovic, unpublished notes.Google Scholar
  46. 46.
    R. Gonczi, Evaluation of the capacity of a strange attractor by a discretization method, University of Nice preprint (1986).Google Scholar
  47. 47.
    P. Grassberger,Phys. Lett. 97A:224 (1983).Google Scholar
  48. 48.
    W. E. Caswell and J. A. Yorke, inDimensions and Entropies in Chaotic Systems, G. Mayer-Kress, ed. (Springer, Berlin, 1986).Google Scholar

Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • P. Grassberger
    • 1
  • R. Badii
    • 2
  • A. Politi
    • 3
  1. 1.Physics DepartmentUniversity of WuppertalWuppertal 1Federal Republic of Germany
  2. 2.Institute of Theoretical PhysicsUniversity of ZurichZurichSwitzerland
  3. 3.Istituto Nazionale di OtticaFirenzeItaly

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