# Uses for Sharkovskii's theorem

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?

• 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. May 13, 2021 at 12:52
• @GerryMyerson You could add a few details about these applications and write that as an answer.
– rimu
May 13, 2021 at 13:50
• OK, I've done that. May 14, 2021 at 4:34
• 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? May 14, 2021 at 7:17
• @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.
– rimu
May 14, 2021 at 15:56

In regard to applications of Sharkovsky's theorem (difference equation models) it is interesting that the result is sturdy to perturbations in $$f$$. Block  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  and Butler and Pianigiani  have related results.