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q-Dimension

D_q=1/(1-q)lim_(epsilon->0)(lnI(q,epsilon))/(ln(1/epsilon),)
(1)

where

I(q,epsilon)=sum_(i=1)^Nmu_i^q,
(2)

epsilon is the box size, and mu_i is the natural measure.

The capacity dimension (a.k.a. box-counting dimension) is given by q=0,

D_0=1/(1-0)lim_(epsilon->0)(ln(sum_(i=1)^(N(epsilon))1))/(-lnepsilon)
(3)
=-lim_(epsilon->0)(ln[N(epsilon)])/(lnepsilon).
(4)

If all mu_is are equal, then the capacity dimension is obtained for any q.

The information dimension corresponds to q=1 and is given by

D_1=lim_(q->1)D_q
(5)
=lim_(q->1)(lim_(epsilon->0)(ln[sum_(i=1)^(N(epsilon))mu_i^q])/(-lnepsilon))/(1-q)
(6)
=lim_(epsilon->0)lim_(q->1)(ln(sum_(i=1)^(N(epsilon))mu_i^q))/((q-1)lnepsilon).
(7)

But for the numerator,

lim_(q->1)ln(sum_(i=1)^(N(epsilon))mu_i^q)=ln(sum_(i=1)^(N(epsilon))mu_i)=ln1=0,
(8)

and for the denominator, lim_(q->1)(q-1)=0, so use l'Hospital's rule to obtain

D_1=lim_(epsilon->0)(1/(lnepsilon)lim_(q->1)(summu_i^qlnmu_i)/1).
(9)

Therefore,

D_1=lim_(epsilon->0)(sum_(i=1)^(N(epsilon))mu_ilnmu_i)/(lnepsilon)
(10)

(Ott 1993, p. 79).

D_2 is called the correlation dimension.

If q_1>q_2, then

D_(q_1)<=D_(q_2)
(11)

(Ott 1993, p. 79).

See also

Capacity Dimension, Correlation Dimension, Fractal Dimension, Information Dimension

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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 D_q 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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q-Dimension

Cite this as:

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

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