Remarks on Sharkovsky's theorem (Q1070619)

When a continuous function has a cycle of given period, it has cycles of other periods specified through the Sharkovsky ordering of the positive integers. In this brief survey, the author states the Sharkovsky theorem and related results for maps of a closed interval, concentrating on simply stated conditions for the existence of certain integer periods. Continuous maps of the circle are briefly discussed and a combinatorial problem associated with the cycle structure of a set of permutations of [1,2,3,...,n] is stated.