Information Dimension

Define the "information function" to be

(1)

where  is the natural measure, or probability that element  is populated, normalized such that

(2)

The information dimension is then defined by

			(3)
			(4)

If every element is equally likely to be visited, then  is independent of , and

(5)

so

(6)

and

			(7)
			(8)
			(9)
			(10)

where  is the capacity dimension.

It satisfies

(11)

where  is the capacity dimension and  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.

Balatoni, J. and Renyi, A. Pub. Math. Inst. Hungarian Acad. Sci. 1, 9, 1956.

Farmer, J. D. "Chaotic Attractors of an Infinite-dimensional Dynamical System." Physica D 4, 366-393, 1982.

Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, p. 79, 1993.

Nayfeh, A. H. and Balachandran, B. Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. New York: Wiley, pp. 545-547, 1995.