q-Dimension
المؤلف:
Grassberger, P
المصدر:
"Generalized Dimensions of Strange Attractors." Phys. Lett. A 97
الجزء والصفحة:
...
25-9-2021
2532
q-Dimension
 |
(1)
|
where
 |
(2)
|
is the box size, and 
is the natural measure.
The capacity dimension (a.k.a. box-counting dimension) is given by 
,
If all 
s are equal, then the capacity dimension is obtained for any 
.
The information dimension corresponds to 
and is given by
But for the numerator,
 |
(8)
|
and for the denominator, 
, so use l'Hospital's rule to obtain
 |
(9)
|
Therefore,
 |
(10)
|
(Ott 1993, p. 79).
is called the correlation dimension.
If 
, then
 |
(11)
|
(Ott 1993, p. 79).
REFERENCES:
Grassberger, P. "Generalized Dimensions of Strange Attractors." Phys. Lett. A 97, 227, 1983.
Hentschel, H. G. E. and Procaccia, I. "The Infinite Number of Generalized Dimensions of Fractals and Strange Attractors." Physica D 8, 435, 1983.
Ott, E. "Measure and the Spectrum of 
Dimensions." §3.3 in Chaos in Dynamical Systems. New York: Cambridge University Press, pp. 78-81, 1993.
Rényi, A. Probability Theory. Amsterdam, Netherlands: North-Holland, 1970.
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