James A. Yorke
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- This article is about the 20th century mathematician. There is also an 18th century clergyman James Yorke
Born in Plainfield, New Jersey, United States, Yorke and his co-author T.Y. Li coined the mathematical term chaos in a paper they published in 1975 entitled "Period Three Implies Chaos", in which it was proved that any continuous 1-dimensional map
- R → R
(1) For each positive integer P, there is a point in R that returns to where it started after P applications of the map and not before. (Of course this means there are infinitely many periodic points, different points for each period P.) This turned out to be a special case of Sharkovsky's theorem.
The second property requires some definitions. A pair of points x and y is called “scrambled” if as the map is applied repeated to the pair, they get closer together and later move apart and then get closer together and move apart, etc, so that they get arbitrarily close together without staying close together. Picture an egg being scrambled forever. You would expect typical atoms x and y to behave in this way. A set S is called "scrambled" if every pair of distinct points in S is scrambled. Scrambling is a kind of mixing.
(2) There is an uncountably infinite set S that is scrambled.
A map satisfying property 2 is sometimes called "chaotic in the sense of Li and Yorke".
References
- T.Y. Li, and J.A. Yorke, Period Three Implies Chaos, Amer. Math. Monthly 82, 985 (1975).
External link
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