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.
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References
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).
R. Benzi, G. Paladin, G. Parisi, and A. Vulpiani,J. Phys. A 17:3521 (1984).
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).
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).
B. B. Mandelbrot,The Fractal Geometry of Nature (Freeman, San Francisco, 1982).
P. Grassberger,Phys. Lett. 97A:227 (1983).
H. G. Hentschel and I. Procaccia,Physica 8D:435 (1983).
P. Grassberger,Phys. Lett. 107A:101 (1985).
V. N. Shtern,Dokl. Akad. Nauk SSSR 270:582 (1983).
J.-P. Eckmann and D. Ruelle,Rev. Mod. Phys. 57:617 (1985).
J. D. Farmer, E. Ott, and J. A. Yorke,Physica 7D:153 (1983).
J.-P. Eckmann and I. Procaccia,Phys. Rev. A 34:659 (1986); G. Paladin, L. Peliti, and A. Vulpiani, University of Rome, Preprint (1986).
V. Jakobson,Commun. Math. Phys. 81:39 (1981); M. Misiurewicz,Publ. Math. IHES 53:17 (1981).
M. Hénon,Commun. Math. Phys. 50:69 (1976).
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).
E. Ott, W. Withers, and J. A. Yorke,J. Stat. Phys. 36:687 (1984).
O. Rössler,Phys. Lett. 57A:397 (1976); P. Holmes,Phil. Trans. R. Soc. A 292:419 (1979).
A. Renyi,Probability Theory (North-Holland, Amsterdam, 1970).
R. Badii and A. Politi,Phys. Rev. Lett. 52:1661 (1984); R. Badii and A. Politi,J. Stat. Phys. 40:725 (1985).
L. P. Kadanoff, private communication.
J. Balatoni and A. Renyi, inSelected Papers of A. Renyi, Vol. 1, p. 558 (Akademia, Budapest, 1976).
S. J. Chang and P. R. Fendlay,Phys. Rev. A 33:4092 (1986).
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.
P. Fredrickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke,J. Diff. Eqs. 49:185 (1983).
P. Grassberger, inChaos in Astrophysics, J. Perdanget al., eds. (Reiedl, Dortrecht, 1985); P. Grassberger, inChaos, A. V. Holden, ed. (Manchester University Press, Manchester, 1986).
P. Grassberger and I. Procaccia,Physica 13D:34 (1984).
P. Billingsley,Ergodic Theory and Information (Wiley, New York, 1965).
F. Takens, Invariants related to dimension and entropy, inAtas do 13° Coloquio Brasileiro de Matematica (1984).
P. Grassberger and I. Procaccia,Physica 9D:189 (1983).
J. D. Farmer, Order within chaos, Thesis, University of California, Santa Cruz (1981).
E. N. Lorenz,Physica 13D:90 (1984);17D:279 (1985).
R. Badii and A. Politi,Phys. Rev. 35A:1288 (1987).
H. Fujisaka,Prog. Theor. Phys. 70:1264 (1983).
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).
B. V. Chirikov and D. L. Shepelyansky,Physica 13D:395 (1984); P. Grassberger and H. Kantz,Phys. Lett. 113A:167 (1985).
P. Grassberger,Physica 14D:365 (1985).
F. Ledrappier and L. S. Young,Ann. Math. 122:509 (1985).
Ya. B. Pesin,Russ. Math. Surv. 32:55 (1977); D. Ruelle,Ann. N. Y. Acad. Sci. 136:229 (1981).
R. Badii and A. Politi,Phys. Scripta 35:243 (1987).
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).
D. Ruelle,Ergod. Theory Dyn. Syst. 2:109 (1982); A. Manning, University of Warwick preprint (1984).
S. Ulam and J. Von Neumann,Bull. Am. Math. Soc. 53:1120 (1947).
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).
S. Roux and C. D. Mitescu,Phys. Rev. B 35:898 (1987).
P. Cvitanovic, unpublished notes.
R. Gonczi, Evaluation of the capacity of a strange attractor by a discretization method, University of Nice preprint (1986).
P. Grassberger,Phys. Lett. 97A:224 (1983).
W. E. Caswell and J. A. Yorke, inDimensions and Entropies in Chaotic Systems, G. Mayer-Kress, ed. (Springer, Berlin, 1986).
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Grassberger, P., Badii, R. & Politi, A. Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors. J Stat Phys 51, 135–178 (1988). https://doi.org/10.1007/BF01015324
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DOI: https://doi.org/10.1007/BF01015324