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 Lebesgue covering dimension are called fractals. The capacity dimension of a compact metric space X is a real number d_(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->0. Explicitly,


(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_(capacity)=d_(information), where d_(information) is the information dimension.

The capacity dimension satisfies


where d_(correlation) is the correlation dimension (correcting the typo in Baker and Gollub 1996).

See also

Correlation Dimension, Correlation Exponent, Dimension, Hausdorff Dimension, Information Dimension, Kaplan-Yorke Dimension

Explore with Wolfram|Alpha


Baker, G. L. and Gollub, J. B. Chaotic Dynamics: An Introduction, 2nd ed. Cambridge, England: Cambridge University Press, 1996.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: Dekker, 1977.

Referenced on Wolfram|Alpha

Capacity Dimension

Cite this as:

Weisstein, Eric W. "Capacity Dimension." From MathWorld--A Wolfram Web Resource.

Subject classifications