In his statistical explorations of experimental results in hydrodynamic turbulence, Mandelbrot (1974) called attention to the need for a multiplicity of characteristic scaling exponents, a range of values for each exponent and their sensitivity to orbital point density distributions (the latter called the Sinai-Ruelle- Bowen or natural measure (Eckmann and Ruelle, 1985)). These needs grew out of the intrinsic heterogeneity in the time dynamics and the nonuniform point distributions in phase space of orbitally divergent, real physical systems. Even with relatively uniform orbital point distributions, it is intuitively obvious that as e > 0, the smaller e- cubes are over-represented and larger e- cubes are under-represented in the M(e) computation (Farmer et al, 1983). For a concrete example, the fraction of the total number of cubes containing say 75% of the points would obviously decrease as the e-lengths studied gets smaller. Normalizing the Dj measures with respect to point densities would correct for this systematic distortion. In addition, the non-systematic influence of real system heterogeneity and non-uniformity in both time and reconstruction space distributions makes the need for relating the Dj
measures to the natural measure even more pressing.
The derivation of many separate scaling exponents, as well as global generalized exponents and the incorporation of point densities in their computation, has been approached by a kind of method of moments (Renyi, 1970; Grassberger, 1983; Hentschel and Procaccia, 1983; Halsey et al, 1986; Mayer-Kress, 1986; Ott et al, 1994). We outline the general arguments here so that the reader will be generally familiar with the ideas and terms, not to serve as a definitive summary. It is a complicated area and the reader will find the required detailed descriptions in
the references. .
We recall that with respect to a statistical distribution, the first moment is the mean; the second moment, o”, the variance; the third moment, o°, the distribution’s asymmetry, the skew; and the fourth moment, o”, its relative peakedness with respect to the probability mass in the tail, called the kurtosis. In these moment computations of an observable x,’s deviation from the mean, |x, — x|*, the value for
q accentuate particular regions of the density distribution. Similarly, the q’s of the
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