By Andre HECK, Jean M. Perdang
Heck A., Perdang J.M., eds making use of fractals in astronomy (Springer, 1991)(ISBN 3540543538)
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Additional resources for Applying fractals in astronomy
Phys. A18, L175. N. (1990): Chaotic Dynamics of Nonlinear Systems (Wiley, New York). J. (1988): Ann. Rev. Astron. Astrophys. 26, 245. , Waldvogel, A. (1986): Phys. Rev. Letters 56,784.
Power law distributions, such as the number of linear oscillation frequencies less than a given frequency, can be interpreted in terms of pseudo-dimensions. We mention here that a second class of power law distributions, such as the number of stars greater than a given mass, the number of impact craters of diameter greater than a given diameter, can be related to the geometry of hierarchical sets. This is made clear by the elementary examples discussed in Perdang (1990b) (first iterate of hierarchy given by the symbol E = E' E", E' a single path, E" a single cycle; the diameters D of the cycle then obey a law N(D) = K D":", a == F.
We finally observe that ODE's and PDE's impose on their solutions a strong mathematical constraint of smoothness in time and in space-time (almost everywhere), a requirement which is absent from CA's. Smoothness of the solution tends to suppress the easy growth of (pre)fractal behaviour which becomes visible only in the limit of very large times (and over very large spatial extensions). In CA's prefractal behaviour is already evident after a small number of timesteps. 4. Fractal Candidates in Astronomy General theoretical arguments are indicative that the following astronomical structures display fractal geometries.