Correlation dimension for iterated function systems

Chin, Wai; Hunt, Brian; Yorke, James

doi:10.1090/S0002-9947-97-01900-4

Correlation dimension for iterated function systems
HTML articles powered by AMS MathViewer

by Wai Chin, Brian Hunt and James A. Yorke PDF

Trans. Amer. Math. Soc. 349 (1997), 1783-1796 Request permission

Abstract:

The correlation dimension of an attractor is a fundamental dynamical invariant that can be computed from a time series. We show that the correlation dimension of the attractor of a class of iterated function systems in $\mathbf {R}^N$ is typically uniquely determined by the contraction rates of the maps which make up the system. When the contraction rates are uniform in each direction, our results imply that for a corresponding class of deterministic systems the information dimension of the attractor is typically equal to its Lyapunov dimension, as conjected by Kaplan and Yorke.

References

J. C. Alexander and J. A. Yorke, Fat baker’s transformations, Ergodic Theory Dynam. Systems 4 (1984), no. 1, 1–23. MR 758890, DOI 10.1017/S0143385700002236
Rufus Bowen and David Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), no. 3, 181–202. MR 380889, DOI 10.1007/BF01389848
Ming Zhou Ding, Celso Grebogi, Edward Ott, Tim Sauer, and James A. Yorke, Estimating correlation dimension from a chaotic time series: when does plateau onset occur?, Phys. D 69 (1993), no. 3-4, 404–424. MR 1251269, DOI 10.1016/0167-2789(93)90103-8
K. J. Falconer, The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 2, 339–350. MR 923687, DOI 10.1017/S0305004100064926
Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
K. J. Falconer, The dimension of self-affine fractals. II, Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 1, 169–179. MR 1131488, DOI 10.1017/S0305004100075253
J. Doyne Farmer, Edward Ott, and James A. Yorke, The dimension of chaotic attractors, Phys. D 7 (1983), no. 1-3, 153–180. MR 719051, DOI 10.1016/0167-2789(83)90125-2
Peter Grassberger, Generalized dimensions of strange attractors, Phys. Lett. A 97 (1983), no. 6, 227–230. MR 718442, DOI 10.1016/0375-9601(83)90753-3
J. S. Geronimo and D. P. Hardin, An exact formula for the measure dimensions associated with a class of piecewise linear maps, Constr. Approx. 5 (1989), no. 1, 89–98. Fractal approximation. MR 982726, DOI 10.1007/BF01889600
Peter Grassberger and Itamar Procaccia, Measuring the strangeness of strange attractors, Phys. D 9 (1983), no. 1-2, 189–208. MR 732572, DOI 10.1016/0167-2789(83)90298-1
H. G. E. Hentschel and Itamar Procaccia, The infinite number of generalized dimensions of fractals and strange attractors, Phys. D 8 (1983), no. 3, 435–444. MR 719636, DOI 10.1016/0167-2789(83)90235-X
James L. Kaplan and James A. Yorke, Chaotic behavior of multidimensional difference equations, Functional differential equations and approximation of fixed points (Proc. Summer School and Conf., Univ. Bonn, Bonn, 1978) Lecture Notes in Math., vol. 730, Springer, Berlin, 1979, pp. 204–227. MR 547989
Ya. B. Pesin, On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions, J. Statist. Phys. 71 (1993), no. 3-4, 529–547. MR 1219021, DOI 10.1007/BF01058436
Y. Pesin and H. Weiss, On the dimension of a general class of deterministic and random Cantor-like sets in $\mathbf {R}^n$, symbolic dynamics, and the Eckmann-Ruelle conjecture, to appear in Comm. Math. Physics.
Mark Pollicott and Howard Weiss, The dimensions of some self-affine limit sets in the plane and hyperbolic sets, J. Statist. Phys. 77 (1994), no. 3-4, 841–866. MR 1301464, DOI 10.1007/BF02179463
A. Rényi, Probability theory, North-Holland Series in Applied Mathematics and Mechanics, Vol. 10, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1970. Translated by László Vekerdi. MR 0315747
David Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math. 34 (1976), no. 3, 231–242. MR 420720, DOI 10.1007/BF01403069
Károly Simon, Hausdorff dimension for noninvertible maps, Ergodic Theory Dynam. Systems 13 (1993), no. 1, 199–212. MR 1213088, DOI 10.1017/S014338570000729X
Overlapping cylinders: the size of dynamically defined Cantor-set, in Ergodic Theory of Z$^d$ actions, London Math. Soc. Lecture Notes 228, Cambridge Univ. Press, 1996.
T. D. Sauer and J. A. Yorke, Are the dimensions of a set and its image equal under typical smooth functions? to appear in Ergod. Th. & Dynam. Sys.
Lai Sang Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems 2 (1982), no. 1, 109–124. MR 684248, DOI 10.1017/s0143385700009615