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Sharkovskii's Theorem
Posted:
Nov 1, 1993 11:52 AM


Sharkovskii's Theorem
There is a theorem which is both beautiful and important in the history of dynamical systems. Thus it deserves its own article. It involves a somewhat strange way to order the integers, stated at the end of the article. I write n ">" k to mean n is greater than k in this strange ordering. Here is the theorem:
Sharkovskii's Theorem (1964): If f is a continuous map from an interval to the real line, then if f has a point of least period n, and if n ">" k, then f has a point of least period k.
This says that just knowing one periodic point can indicate the existence of many other periodic points, without you ever having to find those other points.
In particular, a continuous realvalued map of an interval with a point of period three has points of every other period. This is the subject of one of the most famous papers in dynamical systems, Li and Yorke's "Period three implies chaos," 1975. This paper is the first time that the word chaos was used mathematics; it was a wellreceived paper that provided the name of a new branch of math. Though it turned out that the result in the paper had been proved in stronger form previously (Li and Yorke did not know of Sharkovskii's paper, as it was in an obscure journal in the Soviet Union), the paper stated the results in such an exciting and beautiful way, that it is still quoted to this day.
Strange Ordering of the integers: 3 is "largest," followed by 5 ">" 7 ">" 9 ">" all odd numbers, backwards from the standard ordering of the integers.
Next largest are, 2*3 ">" 2*5 ">" 2*7 ">" in backward order all integers of the form 2 times an odd integer. Then:
4*3 ">" 4*5 ">" 4*7 ">" . . .
. . .
2^n*3 ">" 2^n*5 ">" 2^n*7 ">" . . .
2^(n+1)*3 ">" 2^(n+1)*5 ">" 2^(n+1)*7 ">" . . .
. . .
Finally, . . . 2^n ">" 2^(n1) ">" . . . ">"4 ">" 2 ">" 1.



