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To: Frank Sully who wrote (2270)	11/22/2021 7:17:35 AM
From: Frank Sully	Respond to of 2646

OT: More on Strange Attractors (Continued) During the late 1960’s and early 1970’s, at the University of California at Berkeley, Dr. Smale and his small army of doctoral students, including my thesis advisor Sheldon Newhouse, developed the mathematical theory of stable chaotic dynamical systems, the so-called Axiom A systems. These became the mathematical models for the Lorenz Attractor, the Red Spot model f Jupiter and the onset of turbulence and Strange Attractors. During the late 1970’s and the early 1980’s, Dr. Newhouse established and coordinated a small, elite group of Dynamical Systems researchers, which included Dr. Newhouse, two Ergodic Theorists and two specialists in Foliations. I studied with Dr. Newhouse as my thesis advisor and the Dynamical Systems group from 1977 to 1984, when I received my Ph.D. in the Chaos and Fractal Dimension of Julia Sets in Polynomials of a Complex Variable. This is in the specialty of Ergodic Theory, which studies the randomness and probabilistic behavior of dynamical systems. Julia Sets, besides being very interesting mathematically, are very beautiful and geometrically intricate. Computer explorations of Julia Sets were very popular during the late 1980’s and early 1990’s. Rather than being Strange Attractors, their chaotic sets are Chaotic Repellers: rather than being attracted to the chaotic set in the future, they are repelled by the chaotic set in the future. As fractals, Julia sets are infinitely complex geometrically. The following six minute video is a deep zoom of an interesting Julia set. As a fractal, you can zoom in finer and finer detail and increasing geometrically intricate detail is revealed. (A discussion of the probabilistic models as Bernoulli systems would take us too far afield.) Cheers, Frank BTW, the stock market will open again soon. Back to investments!