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Sharkovskii's theorem is a fascinating result and ready to stand on its own. But is it also used somewhere? Are there other theorems that rely on it or somehow use the Sharkovskii ordering?

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    $\begingroup$ There is a brief discussion of applications of the theorem in Thomas D Rogers, Remarks on Sharkovsky's Theorem, Rocky Mountain J. Math. 15 (1985) 565-569. $\endgroup$ May 13, 2021 at 12:52
  • $\begingroup$ @GerryMyerson You could add a few details about these applications and write that as an answer. $\endgroup$
    – rimu
    May 13, 2021 at 13:50
  • $\begingroup$ OK, I've done that. $\endgroup$ May 14, 2021 at 4:34
  • $\begingroup$ It depends on what you mean by "uses". Sharkovsky's theorem motivated the investigation of "what happens" if there's a point of period 3, from which pretty much all topological chaos theory originated, and that has of course many applications. Does this count? $\endgroup$ May 14, 2021 at 7:17
  • $\begingroup$ @A.DellaCorte Im am more interested in direct influences, not just motivation and inspiration. On the other hand, theorems that use the techniques to prove the theorem and not the statement itself, should count. $\endgroup$
    – rimu
    May 14, 2021 at 15:56

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Quoting from Thomas D Rogers, Remarks on Sharkovsky's Theorem, Rocky Mountain J. Math. 15 (1985) 565-569:

In regard to applications of Sharkovsky's theorem (difference equation models) it is interesting that the result is sturdy to perturbations in $f$. Block [5] shows that if $f$ has a point of period $n$, then there is a neighborhood $N$ of $f$ in $C(I)$ such that for all $g$ in $N$ and all positive integers $k$, with $k$ left of $n$ in the Sharkovsky order, $g$ has a point of period $k$. Kloeden [10] and Butler and Pianigiani [6] have related results.

[5] is L. Block, Stability of periodic orbits in theorem of Sharkovsky, Proc. Amer. Math. Soc. 81 (1981), 333-336.

[10] is P. E. Kloeden, Chaotic difference equations are dense, Bull. Austral. Math. Soc. 15 (1976), 371-379.

[6] is G. J. Butler and G. Pianigiani, Periodic points and chaotic functions in the unit interval, Bull. Austral. Math. Soc. 18 (1978), 255-265.

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