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I recently came across Sharkovskii's Theorem which asserts that if $f:\mathbb{R} \to \mathbb{R}$ is continuous and has a cycle of length $m$, then $f$ has a cycle of length $n$ for any $n$ which comes after $m$ in the following ordering:

$$3 < 5 < 7 < 9 < ...$$ $$ < 2\cdot3 < 2\cdot5 < 2\cdot7 < 2 \cdot 9 < ...$$ $$...$$ $$ < 2^n\cdot3 < 2^n\cdot5 < 2^n\cdot7 < 2^n \cdot 9 < ...$$ $$...$$ $$... < 2^n < 2^{n-1} < ... <2 < 1$$

I was wondering how strongly this theorem depends on the structure of the underlying topological space? For example, the theorem is clearly not true in any discrete space, but it is true for a closed interval. Does anyone know of a fairly general set of a criteria a topological space $X$ must meet for a Sharkovskii-type result to hold for continuous functions $f:X\to X$?

Thanks in advance.

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2 Answers 2

up vote 5 down vote accepted

Sharkovskii's theorem holds on $\mathbb{R}$ and closed intervals $[a,b]\subset\mathbb{R}$ because they are ordered sets, and the order behaves nicely withe respect to the topology. It does not hold however on the circle, which is just a closed interval with the endpoints identified. It has been extended to certain topological spaces that have a linear order. The following is taken from this paper.

Sharkovsky’s Theorem has been generalized in various ways. Schirmer proves that any connected linear order space is a Sharkovsky space [17]. While Schirmer also proves that Theorem 1.2B also holds for connected linear spaces if they contain an arc, Baldwin shows that Theorem 1.2B does not hold for all connected linear spaces [5]. It is interesting to note that Baldwin shows that the only way Theorem 1.2B can fail for a connected linear space is if the space does not admit any map with a certain period. While chainable (or arc-like) continua are generally not Sharkovsky spaces, Minc and Transue show that hereditarily decomposable chainable continua are Sharkovsky spaces [14].

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I suggest you look at a recent paper in Discrete and Continuous Dynamical Systems-A by Conner, Grant and Meilstrup (http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=7409) for more examples of spaces which are Sharkovsky spaces. For example, covering spaces of the Warsaw circle, among others.

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