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

**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*
(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

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1999-05-26