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
If every element is equally likely to be visited, then is independent of , and
|
(5)
|
so
|
(6)
|
and
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).
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.Referenced on Wolfram|Alpha
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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