Sharkovskii's theorem

In mathematics, Sharkovskii's theorem, named after Oleksandr Mikolaiovich Sharkovsky who published it in 1964, is a result about discrete dynamical systems. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period.

The theorem

For some interval , suppose

is a continuous function. We say that the number x is a periodic point of period m if f ^m(x) = x (where f ^m denotes the composition of m copies of f) and having least period m if furthermore f ^k(x) ≠ x for all 0 < k < m. We are interested in the possible periods of periodic points of f. Consider the following ordering of the positive integers:

It consist of:

Every positive integer appears exactly once somewhere on this list. Note that this ordering is not a well-ordering. Sharkovskii's theorem states that if f has a periodic point of least period m and m precedes n in the above ordering, then f has also a periodic point of least period n.