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Sharkovskii's Theorem
Posted:
Nov 1, 1993 11:52 AM
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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 real-valued 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 well-received
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^(n-1) ">" . . . ">"4 ">" 2 ">" 1.
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