Abstract
We consider different definitions of the correlation dimension and find some relationships between them and other characteristics of dimension type such as Hausdorff dimension, box dimension, etc. We also introduce different ways to define and study the generalized spectrum for dimensions—a one-parameter family of characteristics of dimension type.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. Procaccia, P. Grassberger, and V. G. E. Hentschel, On the characterization of chaotic motions, inLecture Notes in Physics, No. 179 (Springer, Berlin, 1983), pp. 212–221.
P. Billingsley,Ergodic Theory and Information (Wiley, New York, 1965).
W. Hurewich and H. Wallman,Dimension Theory (1941).
L.-S. Young, Dimension, entropy and Lyapunov exponents,Ergod. Theory Dynam. Syst. 2:109–124 (1982).
A. Renyi, inDimension, Entropy and Information (Transactions Second Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, 1957), pp. 545–556.
F. Ledrappier and M Misiurewicz, Dimension of invariant measures for maps with exponent zero,Ergod. Theory Dynam. Syst. 5:595–610 (1985).
C. D. Cutler, Connecting ergodicity and dimension in dynamical systems,Ergod. Theory Dynam. Syst. 10:451–462 (1990).
J. P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors,Rev. Mod. Phys. 57:617–653 (1985).
R. Mane,Ergodic Theory and Differential Dynamics (Springer-Verlag, Berlin, 1987).
Ya. B. Pesin, Generalized spectrum for the dimension: The approach based on Carathéodory's construction, inConstantin Carathéodory, an International Tribute, T. M. Rassias, ed. (1991), Vol. 2, pp. 1108–1119.
Ya. B. Pesin, Dimensionlike characteristics for the invariant sets of the dynamical systems,Russ. Math. Surv. 43(4) (1988).
G. E. Hentsel and I. Procaccia,Physica 8D:435 (1983).
P. Frederickson, T. Kaplan, E. Yorke, and G. Yorke, The Lyapunov dimension of strange attractors,J. Differential Equation 49(2):185–206 (1983).
E. Ott, T. Sauer, and G. Yorke, Lyapunov partition functions for the dimensions of chaotic sets,Phys. Rev. A 39(8):4212–4222 (1989).
C. Grebogi, E. Ott, and G. Yorke, Roundoff-induced periodicity and the correlation dimension of chaotic attractors,Phys. Rev. A 38(7):3688–3692 (1988).
T. Farmer, E. Ott, and G. Yorke, The dimension of chaotic attractors,Physica 7D:153–180 (1983).
Ya. Pesin and A. Tempelman, Correlation dimension of group action, preprint, Pennsylvania State University (1992).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pesin, Y.B. On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions. J Stat Phys 71, 529–547 (1993). https://doi.org/10.1007/BF01058436
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01058436