James A. Yorke
||This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (June 2013)|
|James Alan Yorke|
|Born||James Alan Yorke
August 3, 1941
Plainfield, New Jersey
|Fields||Math and Physics (theoretical)|
|Institutions||University of Maryland, College Park|
|Doctoral students||Tien-Yien Li|
Born in Plainfield, New Jersey, United States, Yorke attended The Pingry School, then located in Hillside, New Jersey. In June of 2013, Dr. Yorke retired and stepped down as chair of the University of Maryland's Math department. Yorke is now a Distinguished University Professor Emeritus with the Institute for Physical Science and Technology department at the University of Maryland.
He and Benoit Mandelbrot were the recipients of the 2003 Japan Prize in Science and Technology. Yorke was selected for his work in chaotic systems. In 2012 he became a fellow of the American Mathematical Society.
F:R → R
that has a period-3 orbit must have two properties:
(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 repeatedly 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".
- List of Fellows of the American Mathematical Society, retrieved 2013-09-01.
- A.N. Sharkovskii, Co-existence of cycles of a continuous mapping of the line into itself, Ukrainian Math. J., 16:61-71 (1964)
- T.Y. Li, and J.A. Yorke, Period Three Implies Chaos, American Mathematical Monthly 82, 985 (1975).