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 $D$ equals the Kaplan-Yorke Dimension $D_{KY}$,

$$D = D_{KY} = d_{\rm Lya} = 1+{\sigma_1\over \sigma_2},$$

where $\sigma_1$ and $\sigma_2$ 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 2-D maps, $\nu=\sigma=D$, where $\nu$ is the Correlation Exponent, $\sigma$ is the Information Dimension, and $D$ is the Capacity Dimension (Young 1984).

See also Capacity Dimension, Kaplan-Yorke Dimension, Lyapunov Characteristic Exponent, Lyapunov Dimension

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