Capacity Dimension
المؤلف:
Baker, G. L. and Gollub, J. B.
المصدر:
Chaotic Dynamics: An Introduction, 2nd ed. Cambridge, England: Cambridge University Press, 1996.
الجزء والصفحة:
...
14-9-2021
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Capacity Dimension
A dimension also called the fractal dimension, Hausdorff dimension, and Hausdorff-Besicovitch dimension in which nonintegral values are permitted. Objects whose capacity dimension is different from their Lebesgue covering dimension are called fractals. The capacity dimension of a compact metric space 
is a real number 
such that if 
denotes the minimum number of open sets of diameter less than or equal to 
, then 
is proportional to 
as 
. Explicitly,
(if the limit exists), where 
is the number of elements forming a finite cover of the relevant metric space and 
is a bound on the diameter of the sets involved (informally, 
is the size of each element used to cover the set, which is taken to approach 0). If each element of a fractal is equally likely to be visited, then 
, where 
is the information dimension.
The capacity dimension satisfies
where 
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.
Nayfeh, A. H. and Balachandran, B. Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. New York: Wiley, pp. 538-541, 1995.
Peitgen, H.-O. and Richter, D. H. The Beauty of Fractals: Images of Complex Dynamical Systems. New York: Springer-Verlag, 1986.
Wheeden, R. L. and Zygmund, A. Measure and Integral: An Introduction to Real Analysis. New York: Dekker, 1977.
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