q-Dimension

(1)

where

(2)

 is the box size, and  is the natural measure.

The capacity dimension (a.k.a. box-counting dimension) is given by ,

			(3)
			(4)

If all s are equal, then the capacity dimension is obtained for any .

The information dimension corresponds to  and is given by

			(5)
			(6)
			(7)

But for the numerator,

(8)

and for the denominator, , so use l'Hospital's rule to obtain

(9)

Therefore,

(10)

(Ott 1993, p. 79).

 is called the correlation dimension.

If , then

(11)

(Ott 1993, p. 79).

REFERENCES:

Grassberger, P. "Generalized Dimensions of Strange Attractors." Phys. Lett. A 97, 227, 1983.

Hentschel, H. G. E. and Procaccia, I. "The Infinite Number of Generalized Dimensions of Fractals and Strange Attractors." Physica D 8, 435, 1983.

Ott, E. "Measure and the Spectrum of  Dimensions." §3.3 in Chaos in Dynamical Systems. New York: Cambridge University Press, pp. 78-81, 1993.

Rényi, A. Probability Theory. Amsterdam, Netherlands: North-Holland, 1970.