Capacity Dimension

A Dimension also called the Fractal Dimension, Hausdorff Dimension, and Hausdorff-Besicovitch Dimension in which nonintegral values are permitted. Objects whose capacity dimension is different from their Topological Dimension are called Fractals. The capacity dimension of a compact Metric Space $X$ is a Real Number $d_{\rm capacity}$ such that if $n(\epsilon)$ denotes the minimum number of open sets of diameter less than or equal to $\epsilon$, then $n(\epsilon)$ is proportional to $\epsilon^{-D}$ as $\epsilon\to 0$. Explicitly,

$$d_{\rm capacity} \equiv - \lim_{\epsilon \to 0^+} {\ln N \over \ln \epsilon}$$

(if the limit exists), where $N$ is the number of elements forming a finite Cover of the relevant Metric Space and $\epsilon$ is a bound on the diameter of the sets involved (informally, $\epsilon$ is the size of each element used to cover the set, which is taken to approach 0). If each element of a Fractal is equally likely to be visited, then $d_{\rm capacity} = d_{\rm information}$, where $d_{\rm information}$ is the Information Dimension. The capacity dimension satisfies

$$d_{\rm correlation} \leq d_{\rm information} \leq d_{\rm capacity}$$

where $d_{\rm correlation}$ is the Correlation Dimension, and is conjectured to be equal to the Lyapunov Dimension.

See also Correlation Exponent, Dimension, Hausdorff Dimension, Kaplan-Yorke Dimension

References

Nayfeh, A. H. and Balachandran, B. Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. New York: Wiley, pp. 538-541, 1995.

Peitgen, H.-O. and Richter, D. H. The Beauty of Fractals: Images of Complex Dynamical Systems. New York: Springer-Verlag, 1986.

Wheeden, R. L. and Zygmund, A. Measure and Integral: An Introduction to Real Analysis. New York: M. Dekker, 1977.