“generalized dimensions,” Dg, emphasize different aspects of the relative point
density that are assumed to be uniform in the computation of Do. We recall from
above that the power law slope constituting D, = lim a 7 é>0 sal é
. If we emphasize the
component of the probability (measure, uw) or, equivalently, time spent by the orbit in
cube i, “(C,) instead of simply the number of cubes occupied by any points, M(e),
along with the different length scales of the cube as e—0 we have a generalized dimension. A common expression for the generalized dimension includes the
fractional pre-factor in q written so as to make things come out right:
: M(e) D, = ,__finn sel where /(qg,¢) = SLAC I. The higher the q, the greater q-le—>0 In(e) =
the dominance of the higher probability cubes, ~(C,). To see how this q-induced
separation in emphasis might work, if the ratio for q = 2 between the probability containing cubes 0.25 and 0.05 is 25, their ratio for q = 3 is 125. For q = 0, the scaling exponent is the capacity dimension. This result of the actions of a changing q has been analogized to the way changing temperature in a thermodynamic
system evokes different aspects of its behavior.
The “multifractal formalism” generally begins by determining the statistical densities over a range of scale lengths by one means or another including wavelet transformations across wavelength scale (Arneodo et al, 1988). These densities by scale are then systematically raised to a range of q exponents. Since q, and therefore Dg, can vary continuously, functions are created that shows how D, varies with q. These are then further transformed, resulting in a single maximum parabolic curve whose shape and size is sensitive to the conditions of the experiment (Halsey et al, 1986). Generalized dimensions decrease as q increases. A unique neuropsychopharmacological application of the multifractal technique to a study of the behavioral influence of increasing amounts of cocaine on the time-dependent patterns of spatial exploration, temporal-spatial fluctuations, in rats, demonstrated a global splitting in the parabolic distribution suggestive of a cocaine-induced global
phase transition, not unlike the well-known, dose-dependent, amphetamine-induced
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