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  is a real number  such that if  denotes the minimum number of open sets of diameter less than or equal to , then  is proportional to  as . Explicitly,

(if the limit exists), where  is the number of elements forming a finite cover of the relevant metric space and  is a bound on the diameter of the sets involved (informally,  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 , where  is the information dimension.

The capacity dimension satisfies

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

REFERENCES:

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.