Sharkovskii's theorem

In mathematics, Sharkovskii's theorem, named after Oleksandr Mykolaiovych Sharkovskii 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.

For some interval ${\displaystyle I\subset \mathbb {R} }$, 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 consists 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.

As a consequence, we see that if f has only finitely many periodic points, then they must all have periods which are powers of two. Furthermore, if there is a periodic point of period three, then there are periodic points of all other periods.

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