Kaplan-Yorke Conjecture

There are several versions of the Kaplan-Yorke conjecture, with many of the higher dimensional ones remaining unsettled. The original Kaplan-Yorke conjecture (Kaplan and Yorke 1979) proposed that, for a two-dimensional mapping, the capacity dimension  equals the Kaplan-Yorke dimension ,

where  and  are the Lyapunov characteristic exponents. This was subsequently proven to be true in 1982. A later conjecture held that the Kaplan-Yorke dimension is generically equal to a probabilistic dimension which appears to be identical to the information dimension (Frederickson et al. 1983). This conjecture is partially verified by Ledrappier (1981). For invertible two-dimensional maps, , where  is the correlation exponent,  is the information dimension, and  is the capacity dimension (Young 1984).

REFERENCES:

Chen, Z. M. "A Note on Kaplan-Yorke-Type Estimates on the Fractal Dimension of Chaotic Attractors." Chaos, Solitons, and Fractals 3, 575-582, 1994.

Frederickson, P.; Kaplan, J. L.; Yorke, E. D.; and Yorke, J. A. "The Liapunov Dimension of Strange Attractors." J. Diff. Eq. 49, 185-207, 1983.

Kaplan, J. L. and Yorke, J. A. In Functional Differential Equations and Approximations of Fixed Points: Proceedings, Bonn, July 1978 (Ed. H.-O. Peitgen and H.-O. Walther). Berlin: Springer-Verlag, p. 204, 1979.

Ledrappier, F. "Some Relations Between Dimension and Lyapunov Exponents." Commun. Math. Phys. 81, 229-238, 1981.

Worzbusekros, A. "Remark on a Conjecture of Kaplan and Yorke." Proc. Amer. Math. Soc. 85, 381-382, 1982.

Young, L. S. "Dimension, Entropy, and Lyapunov Exponents in Differentiable Dynamical Systems." Phys. A 124, 639-645, 1984.