Abstract
The probability measure generated by typical chaotic orbits of a dynamical system can have an arbitrarily fine-scaled interwoven structure of points with different singularity scalings. Recent work has characterized such measures via a spectrum of fractal dimension values. In this paper we pursue the idea that the infinite number of unstable periodic orbits embedded in the support of the measure provides the key to an understanding of the structure of the subsets with different singularity scalings. In particular, a formulation relating the spectrum of dimensions to unstable periodic orbits is presented for hyperbolic maps of arbitrary dimensionality. Both chaotic attractors and Chaotic repellers are considered.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
D. Farmer, E. Ott, and J. A. Yorke, Physica D 7, 153 (1983)
Grassberger, Phys. Lett. 97A, 227 (1983).
G. E. Hentschel and I. Procaccia, Physica D 8, 435 (1983).
P. Grassberger, Phys. Lett. 107A, 101 (1985); T. C. Halsey, M.J. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman,Phys. Rev. A 33, 1141 (1986).
C. Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. A 36, 3522 (1987). Recent related works are the following: T. Morita, H. Hata, H. Mori, T. Horita, nd K. Tornita, Progr. Theor. Phys. 78, 511 (1987); G. Gunaratne and I. Procaccia, Phys. Rev. Lett. 59, 1377 ( 1987 ); M. H. Jensen (unpublished).
Bowen, Trans. Am. Math. Soc. 154, 377 (1971).
B. Katok, Publ. Math. IHES 51, 137 (1980).
P. Kadanoff and C. Tang, Proc. Natl. Acad. Sci. USA 81, 1276 (1984).
Auerbach, P. Cvitanovic, J.-P. Eckmann, G. Gunaratne, and I. Procaccia, Phys. Rev. Lett. 58, 2387 (1987).
Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. Lett. 57, 1284 (1986); Phys. Rev. A 36, 5365 (1987).
V. Berry, in Chaotic Behavior of Deterministic Systems, edited by G. Iooss, R. H. G. Heileman, and R. Stora ( North-Holland, Amsterdam, 1983 ), p. 171–271.
H. Hannay and A. M. Ozorio de Almeida, J. Phys. A 17, 3429 (1984).
V. Berry, Proc. R. Soc. London A 400, 229 (1985).
F. Hausdorff, Math. Annalen 79, 157 (1918).
N. Kolmogorov, Dokl. Akad. Nauk SSSR 119, 861 (1958).
A. Renyi, Acta Mathematica (Hungary) 10, 193 (1959).
Kaplan and J. Yorke, in Functional Differential Equations and the Approximation of Fixed Points, Vol. 730 of Lecture Notes in Mathematics, edited by H. O. Peitgen and H. O. Walther ( Springer, Berlin, 1978 ), p. 228.
R. Bowen, On Axiom A Diffeomorphisms,CBMS Regional Conference Series in Mathematics (American Mathematical Society, Providence, 1978), Vol. 35.
Grebogi, E. Ott, and J. A. Yorke, Physica D 7, 181 (1983).
H. Kantz and P. Grassberger, Physica D 17, 75 (1985); G. Hsu, E. Ott, and C. Grebogi, Phys. Lett. A (to be published).
P. Szépfalusy and T. Tél, Phys. Rev. A 34, 2520 (1986).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1988 Springer Science+Business Media New York
About this chapter
Cite this chapter
Grebogi, C., Ott, E., Yorke, J.A. (1988). Unstable periodic orbits and the dimensions of multifractal chaotic attractors. In: Hunt, B.R., Li, TY., Kennedy, J.A., Nusse, H.E. (eds) The Theory of Chaotic Attractors. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21830-4_19
Download citation
DOI: https://doi.org/10.1007/978-0-387-21830-4_19
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-2330-1
Online ISBN: 978-0-387-21830-4
eBook Packages: Springer Book Archive