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Correlation dimension for iterated function systems

Author(s): Wai Chin; Brian Hunt; James A. Yorke
Journal: Trans. Amer. Math. Soc. 349 (1997), 1783-1796.
MSC (1991): Primary 28D20, 28D05; Secondary 60G18
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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:

[AY]
J. C. Alexander and J. A. Yorke, Fat baker's transformations, Ergod. Th. & Dynam. Sys. 4 (1984), 1-23. MR 86c:58090

[BR]
R. Bowen and D. Ruelle, The ergodic theory of Axiom-A flows, Invent. Math. 29 (1975), 181-202. MR 52:1786

[DGOSY]
M. Ding, C. Grebogi, E. Ott, T. Sauer, and J. A. Yorke, Estimating correlation dimension from a chaotic time series: when does plateau onset occur? Physica D 69 (1993), 404-424. MR 94g:58136

[F1]
K. J. Falconer, The Hausdorff dimension of self-affine fractals, Math. Proc. Camb. Phil. Soc. 103 (1988), 339-350. MR 89h:28010

[F2]
K. J. Falconer, Fractal Geometry, Mathematical Foundations and Applications, John Wiley & Sons, 1990. MR 92j:28008

[F3]
K. J. Falconer, The dimension of self-affine fractals II, Math. Proc. Camb. Phil. Soc. 111 (1992), 169-179. MR 92m:28010

[FOY]
J. D. Farmer, E. Ott, and J. A. Yorke, The dimension of chaotic attractors, Physica D 7 (1983), 153-180. MR 84m:58022

[G]
P. Grassberger, Generalized dimensions of strange attractors, Physics Letters A 97 (1983) 227. MR 84i:58075

[GH]
J. S. Geronimo and D. P. Hardin, An exact formula for the measure dimensions associated with a class of piecewise linear maps, Constructive Approximation 5 (1989), 89-98. MR 90d:58076

[GP]
P. Grassberger and I. Procaccia, Characterization of strange attractors, Phys. Rev. Lett. 50 (1983) 346; Measuring the strangeness of strange attractors, Physica D 9 (1983) 189. MR 85i:58071

[HP]
H. G. E. Hentschel and I. Procaccia, The infinite number of generalized dimensions of fractals and strange attractors, Physica D 8 (1983), 435-444. MR 85a:58064

[KY]
J. L. Kaplan and J. A. Yorke, Chaotic behavior of multidimensional difference equations, in Functional Differential Equations and Approximations of Fixed Points, edited by H.-O. Peitgen and H.-O. Walter, Lecture Notes in Mathematics 730, Springer, Berlin, p.204. MR 80k:58074

[P]
Y. B. Pesin, On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions, Journal of Statistical Physics 71 (1993), 529-547. MR 94d:28008

[PW]
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.

[PoW]
M. Pollicott and H. Weiss, The dimensions of some self-affine limit sets in the plane and hyperbolic sets, Journal of Statistical Physics 77 (1994), 841-866. MR 95h:58083

[R]
A. Renyi, Probability Theory, North-Holland, Amsterdam, 1970. MR 47:4296

[Ru]
D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math. 98 (1976), 619-654. MR 54:8732

[S1]
K. Simon, Hausdorff dimension for non-invertible maps, Ergod. Th. & Dynam. Sys. 13 (1993), 199-212. MR 94c:58146

[S2]
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.

[SY]
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.

[Y]
L. S. Young, Dimension, entropy and Lyapunov exponents, Ergod. Th. & Dynam. Sys. 2 (1982), 109-124. MR 84h:58087

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Additional Information:

Wai Chin
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523 (On leave at: Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, Minnesota 55455)
Email: chin@ima.umn.edu

Brian Hunt
Affiliation: Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742
Email: bhunt@ipst.umd.edu

James A. Yorke
Affiliation: Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742
Email: yorke@ipst.umd.edu

DOI: 10.1090/S0002-9947-97-01900-4
PII: S 0002-9947(97)01900-4
Received by editor(s): June 30, 1995
Copyright of article: Copyright 1997, American Mathematical Society

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