Order the Natural Numbers as follows:

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 for every Integer .

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

**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.

© 1996-9

1999-05-26