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Information Dimension

Define the "information function" to be

I=-sum_(i=1)^NP_i(epsilon)ln[P_i(epsilon)],
(1)

where P_i(epsilon) is the natural measure, or probability that element i is populated, normalized such that

sum_(i=1)^NP_i(epsilon)=1.
(2)

The information dimension is then defined by

d_(inf)=-lim_(epsilon->0^+)I/(ln(epsilon))
(3)
=lim_(epsilon->0^+)sum_(i=1)^(N)(P_i(epsilon)ln[P_i(epsilon)])/(ln(epsilon)).
(4)

If every element is equally likely to be visited, then P_i(epsilon) is independent of i, and

sum_(i=1)^NP_i(epsilon)=NP_i(epsilon)=1,
(5)

so

P_i(epsilon)=1/N,
(6)

and

d_(inf)=lim_(epsilon->0^+)(sum_(i=1)^N1/Nln(1/N))/(lnepsilon)
(7)
=lim_(epsilon->0^+)(ln(N^(-1)))/(lnepsilon)
(8)
=-lim_(epsilon->0^+)(lnN)/(ln(epsilon))
(9)
=d_(cap),
(10)

where d_(cap) is the capacity dimension.

It satisfies

d_(correlation)<=d_(information)<=d_(capacity)
(11)

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

See also

Capacity Dimension, Correlation Dimension, Correlation Exponent

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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.

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Information Dimension

Cite this as:

Weisstein, Eric W. "Information Dimension." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InformationDimension.html

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