Sarkovskii's Theorem

Order the Natural Numbers as follows:

$3\prec 5\prec 7\prec 9\prec 11\prec 13\prec 15\prec\ldots\prec 2\cdot 3\prec 2\cdot 5\prec 2\cdot 7$
$\prec 2\cdot 9\prec\ldots\prec 2\cdot 2\cdot 3\prec 2\cdot 2\cdot 5\prec 2\cdot 2\cdot 7$
$\prec 2\cdot 2\cdot 9\prec\ldots\prec 2\cdot 2\cdot 2\cdot 3\prec\ldots\prec 2^5\prec 2^4\prec 2^3\prec 2^2\prec 2\prec 1.$

Now let $F$ be a Continuous Function from the Reals to the Reals and suppose $p\prec q$ in the above ordering. Then if $F$ has a point of Least Period $p$, then $F$ also has a point of Least Period $q$.

A special case of this general result, also known as Sarkovskii's theorem, states that if a Continuous Real function has a Periodic Point with period 3, then there is a Periodic Point of period $n$ for every Integer $n$.

A converse to Sarkovskii's theorem says that if $p\prec q$ in the above ordering, then we can find a Continuous Function which has a point of Least Period $q$, but does not have any points of Least Period $p$ (Elaydi 1996). For example, there is a Continuous Function with no points of Least Period 3 but having points of all other Least Periods.

See also Least Period

References

Conway, J. H. and Guy, R. K. ``Periodic Points.'' In The Book of Numbers. New York: Springer-Verlag, pp. 207-208, 1996.

Devaney, R. L. An Introduction to Chaotic Dynamical Systems, 2nd ed. Reading, MA: Addison-Wesley, 1989.

Elaydi, S. ``On a Converse of Sharkovsky's Theorem.'' Amer. Math. Monthly 103, 386-392, 1996.

Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, p. 49, 1993.

Sharkovsky, A. N. ``Co-Existence of Cycles of a Continuous Mapping of a Line onto Itself.'' Ukranian Math. Z. 16, 61-71, 1964.

Stefan, P. ``A Theorem of Sharkovsky on the Existence of Periodic Orbits of Continuous Endomorphisms of the Real Line.'' Comm. Math. Phys. 54, 237-248, 1977.