The February 2007 issue of Americam Mathematical Monthly has another short proof of Sharkowski's (Sakowskii's) Theorem. We all know that a continuous map of the closed interval I=[0,1] to itself has a fixed point, that is number x in [0,1] with f(x)=x ( this is one version of Intermediate Value Theorem). Since f(x) ia also in I, we can apply f to f(x) and keep interating . Call the nth interate f^n (x). If f^n (x)=x but not for a smaller integer, then x is called a periodic point of order n for f. The Sharkowski Theorem gives a strange order on the natural numbers asserts that if f has a periodic point of order n, then f has a periodic point of order m for every m less than n in the Sharkowski order. The Sharkowski order is the following: First take odd numbers in the reverse order 3>5>7>.., the multiples of these numbers by 2 in the reverse order 2.3>2.5>2.7>..., then 4.3>4.5>4.7>...>8.3>8.5>8.7>...16.3>16.5>...and finally all powers of two in the usual order 32>16>8>4>2>1. The number one is considered the zeroth power of 2.
That is the strange theorem; I came across it in 80's when some economists wanted me to explain the proofs. Most proofs are elementary but the above seems to be very short (the author thanks Sharkowski for suggestions). The theorem in particular asserts that if f has a periodic point of order 3, then it has a periodic point of every order. This corollary became famous as the statement "period three implies chaos". Some connections are explained in the Wikipedia article http://en.wikipedia.org/wiki/Sarkovskii's_theorem. Another generalization of the above version of the Intermediate Value Theorem is the Brouwer Fixed Point Theorem. This again has a nice constructive proof (which finds approximate fixed points) by economists Scarf and Eves.
