Chaotic Dynamics and Fractals by Michael F. Barnsley, Stephen G. Demko

By Michael F. Barnsley, Stephen G. Demko

1986 1st Ed. Vol. 2 educational Press

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There is a unique set A c K such that A = N u W n (A) . n=l n Moreover, with probability one, A can be found by the coin toss algorithm. Notice how A is a union of continuously altered shrunken copies of itself, which shows why A is fractal in spirit. By the last statement in the Theorem we mean this. ),W2(x ,) ,. ) . Then JC JC n~i with probability one A = {Lim x }.

55 More generally, properties of these objects are expressed in the language of analysis using various types of dimension, connectivity, accessibility, etc. However, a com­ mon language which conveys their geometry is missing: if I wish to describe my fractal to a colleague elsewhere, 1 had best send him a sequences of pictures since usually the diverse languages do not suffice. process. In some cases I can abbreviate the Instead of pictures I send the algorithm; for example, to convey a Julia set, send the corresponding rational map, and to study the set — study the map.

Value this gift? What Hark to the answer given by the Nietzsche in his marvelously prescient benediction: Yea verily, I say unto you: A man must have Chaos yet within him To birth a dancing star. 52 Joseph Ford REFERENCES 1. Michael Berry, "Regular and Irregular Motion," in Topics in Nonlinear Dynamics, AIP Conference Proceedings, Vol. 46 (Am. Inst. , New York, 1978). 2. Robert H. G. Helleman, "Self-Generated Chaotic Behavior in Nonlinear Mechanics," in Fundamental Problems in Statistical Mechanics, Vol.

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