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Sharkovsky's Theorem

Order the natural numbers as follows:

3≺5≺7≺9≺11≺13≺15≺...≺2·3≺2·5≺2·7 ≺2·9≺...≺2·2·3≺2·2·5≺2·2·7 ≺2·2·9≺...≺2·2·2·3≺...≺2^5≺2^4≺2^3≺2^2≺2≺1.

Now let F be a continuous function from the reals to the reals and suppose p≺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 Sharkovsky'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 Sharkovsky's theorem says that if p≺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.

Sharkovsky's theorem includes the period three theorem as a special case (Borwein and Bailey 2003, p. 79).

See also

Least Period, Mann Iteration, Period Three Theorem

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References

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.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.S_arkov'skiĭ, O. M. "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.

Referenced on Wolfram|Alpha

Sharkovsky's Theorem

Cite this as:

Weisstein, Eric W. "Sharkovsky's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SharkovskysTheorem.html

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