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Information Dimension

Define the ``information function'' to be

I \equiv -\sum_{i=1}^{N} P_i(\epsilon)\ln[P_i(\epsilon)],
\end{displaymath} (1)

where $P_i(\epsilon)$ is the Natural Measure, or probability that element $i$ is populated, normalized such that
\sum_{i=1}^N P_i(\epsilon) = 1.
\end{displaymath} (2)

The information dimension is then defined by
$\displaystyle d_{\rm inf}$ $\textstyle \equiv$ $\displaystyle - \lim_{\epsilon\to 0^+} {I\over\ln(\epsilon)}$  
  $\textstyle =$ $\displaystyle \lim_{\epsilon\to 0^+} \sum_{i=1}^{N} {P_i(\epsilon)\ln[P_i(\epsilon)]\over\ln(\epsilon)}.$ (3)

If every element is equally likely to be visited, then $P_i(\epsilon)$ is independent of $i$, and
\sum_{i=1}^N P_i(\epsilon) = N P_i(\epsilon) = 1,
\end{displaymath} (4)

P_i(\epsilon) = {1\over N},
\end{displaymath} (5)

$\displaystyle d_{\rm inf}$ $\textstyle =$ $\displaystyle \lim_{\epsilon\to 0^+} {\sum\limits_{i=1}^N {1\over N}\ln\left({1\over N}\right)\over\ln\epsilon}$  
  $\textstyle =$ $\displaystyle \lim_{\epsilon\to 0^+} {\ln(N^{-1})\over\ln \epsilon}
= -\lim_{\epsilon\to 0^+} {\ln N\over\ln(\epsilon)} = d_{\rm cap},$ (6)

where $d_{\rm cap}$ is the Capacity Dimension.

See also Correlation Exponent


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

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

© 1996-9 Eric W. Weisstein