A SIMPLE SPECIAL CASE OF SHARKOVSKII’S THEOREM

Reid Barton and Keith Burns

In this note I ⊂ ℝ is a bounded closed interval and f : I → I will be a continuous map; f^{n} denotes the n-fold composition of f with itself. A point x ∈ I is a periodic point for f with period p if f^{p}(x) = x and has least period p if in addition f^{k}(x) ≠ x for 1 ≤ k ≤ p−1. The note presents a short proof of the following result.

Proposition. If f has a periodic point that is not fixed, then f has a periodic point of least period 2.

This is a special case of Sharkovskii’s famous theorem, which states that if f has a periodic point with least period p and q comes after p in the ordering

3, 5, 7, …, 2 · 3, 2 · 5, 2 · 7, …, 2^{2} · 3, 2^{2} · 5, 2^{2} · 7, …, 2^{3}, 2^{2}, 2, 1,

then f has a periodic point with least period q.

The proposition corresponds to the fact that 2 is the penultimate number in this ordering. The other simple special case of Sharkovskii’s theorem, namely that an orbit of least period 3 forces the existence of an orbit of least period q for any q, corresponds to the fact that 3 is the first number in the ordering. It was presented in this journal by Li and Yorke [6], who rediscovered it independently of Sharkovskii’s work.

According to the history given in [1], the proposition was Sharkovskii’s first step towards his theorem. Coppel [3] proved in the 1950’s that f^{n}(x) converges as n → ∞ for all x ∈ I if and only if f has no periodic points with least period 2. In 1960 Sharkovskii [8] reproved Coppel’s result and observed that it implies the proposition — because f^{n}(x) does not converge if x is a periodic point whose least period is 2 or more. Sharkovskii completed the proof of his theorem in two subsequent papers [9, 10].

Proof of the Proposition. We prove that if f has a periodic point with least period p > 2, then f has a periodic point that is not fixed and has least period less than p. A descending induction then shows that there is a periodic point with least period 2.

Let x_{1} < x_{2} < ⋯ < x_{p} be the points on an orbit of least period p. We consider the directed graph with vertices 1, …, p − 1 in which vertex i is joined to vertex j if and only if f([x_{i}, x_{i+1}]) ⊃ [x_{j}, x_{j+1}]. Each vertex i must be joined to at least one vertex j ≠ i, because otherwise f would have to permute the endpoints of [x_{i}, x_{i+1}], which is impossible since these points lie on a periodic orbit for f with least period p ≥ 3.

Starting at the vertex 1, choose an edge that joins 1 to a different vertex. Then we join this vertex to a different vertex, and so on. The path can be extended indefinitely with

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each edge joining two different vertices. After at most p − 1 vertices it must return to a previously visited vertex. This gives us a loop i_{1}, …, i_{q} that passes through at least 2 and at most p − 1 vertices.

Set I_{k} = [x_{i_{k}}, x_{i_{k}+1}] for k = 1, …, q. Then f(I_{k}) ⊃ I_{k+1} for k = 1, …, q − 1 and f(I_{q}) ⊃ I_{1}. There is a closed subinterval I′_{1} ⊂ I_{1} such that f(I′_{1}) = I_{2}. In order to see this, look at the intersection of the graph of f with the rectangle I_{1} × I_{2}. At least one component of this intersection must join the top and bottom edges of I_{1} × I_{2}. The interval I′_{1} is the projection to I_{1} of such a component.

Since f^{q}(I′_{1}) = f^{q−1}(f(I′_{1})) = f^{q−1}(I_{2}) ⊃ I_{1} ⊃ I′_{1}, it follows from the intermediate value theorem that f^{q} has a fixed point z in I′_{1}. Clearly z is a periodic point for f whose least period is a factor of q and therefore less than p.

We now show that z is not a fixed point for f. Since z ∈ I′_{1} ⊂ I_{1} and f(z) ∈ f(I′_{1}) = I_{2}, we can have z = f(z) only if z ∈ I_{1} ∩ I_{2}. But I_{1} and I_{2} have disjoint interiors and their endpoints belong to an orbit with least period p > 1. □

The modern proof of the full Sharkovskii theorem is more intricate than the above argument, but does not require any more sophisticated tools. It was first given in [2] and can be found in many texts on dynamical systems, for example [1, 4, 5, 7]. One of the steps is a special case of our proposition, namely that f has a periodic point with least period 2 if f has a periodic point with even least period. The argument presented here is simpler than the standard proof of this step.

ACKNOWLEDGEMENTS. The first author thanks Boris Hasselblatt for his stimulating course in dynamical systems, which introduced him to Sharkovskii’s theorem. The second author thanks the National Science Foundation for support through grant DMS 9803346. We thank Boris Hasselblatt, Amie Wilkinson and the referee for several suggestions that improved this note.

References

[1] L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, World Scientific Publishing Co., 1993.

[2] L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one dimensional maps, pp. 18–34 in Global Theory of Dynamical Systems, Z. Nitecki and C. Robinson eds, Lecture Notes in Mathematics vol. 819, Springer Verlag, 1980.

[3] W. A. Coppel, The solution of equations by iteration, Proc. Camb. Phil. Soc. 51 (1955), 41–43.

[4] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, 1989.

[5] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encylopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, 1995.

[6] T. Li and J. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985–992.

[7] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos, CRC Press, 1995.

[8] A. N. Sharkovskii, Necessary and sufficient conditions for convergence of one-dimensional iteration processes, Ukrain. Math. Zh. 12 (1960), 484–489.

[9] A. N. Sharkovskii, On the reducibility of a continuous function of a real variable and the structure of the stationary points of the corresponding iteration process, Soviet Math. Dokl. 2 (1961), 1062–1064.

[10] A. N. Sharkovskii, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Math. Zh. 16 (1964), 61–71.

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