Talk abstract:

Period Three, Chaos, and the Golden Mean Shift

Danrun Huang
Department of Mathematics
St. Cloud State University
DHuang@stcloudstate.edu

Joint work with Daniel Scully.

In 1975, Li and Yorke first introduced the term "chaos" into scientific literature in the now-famous article "Period Three Implies Chaos". One well-known example of period 3 is a simple piece-wise function defined on the unit interval that has been studied and referred to in many papers, including very recent ones. In this article, we demonstrate the power and beauty of Symbolic Dynamics and Coding by digitizing this same function through the famous Golden Mean Shift in symbolic dynamics. With this approach, we are able to disclose many dynamical properties (we believe new) of this function. In particular, we count and locate all the period-n points of the function. We also provide the complete algorithms of encoding and decoding, including the one for the ambiguous boundary of the Markov Partition. The process is related to the Kneading Theory first introduced by Milnor and Thurston in 1980's.

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1998-1999 Mathematics in Biology