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To: Frank Sully who wrote (1550)	11/1/2021 6:34:09 PM
From: Frank Sully	Respond to of 5488

My Mathematical Heritage (Continued) The modern mathematical field of differential dynamical systems was pioneered by my mathematical grandfather Steve Smale at The University of California at Berkeley during the 1960’s. He had many Ph.D. students who developed the field of differentiable dynamics including my thesis advisor, Sheldon Newhouse, who directed my Ph.D. on randomness and dimension of Julia sets in complex polynomials. Smale’s first major insight of chaos in dynamical systems was the Horseshoe. He discovered this on the beach in Rio de Janiero on a research leave. The following thirteen minute video gives a good introduction to dynamical systems, the horseshoe and coding the horseshoe as a Bernoulli system, the fair coin toss.