Vol. 96, No. 2, 1981

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A graph-theoretic proof of Sharkovsky’s theorem on the periodic points of continuous functions

Chung-Wu Ho and Charles E. Morris, Jr.

Vol. 96 (1981), No. 2, 361–370

Let f be a continuous real valued function defined on the real line. If f has a periodic point of period k, does f have to have a periodic point of some other period m? A Russian mathematician, A. N. Sharkovsky obtained a complete answer to this question. Sharkovsky’s result is elegant, however, his proof is difficult. Recently, P. D. Straffin attempted to give a simple proof of the sufficient part of Sharkovsky’s theorem by means of directed graphs. However, his proof contains a gap. In this paper, the authors fill in the gap in Straffin’s work. They also give a proof of the necessary part of the theorem, which is also based on directed graphs, and thus, obtain a complete simple proof of Sharkovsky’s theorem.

Mathematical Subject Classification 2000
Primary: 58F20
Secondary: 05C20, 92A15
Received: 8 February 1980
Revised: 19 May 1980
Published: 1 October 1981
Chung-Wu Ho
Charles E. Morris, Jr.