[Sharkovskii's theorem](https://en.wikipedia.org/wiki/Sharkovskii%27s_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?



Uses for Sharkovskii's theorem
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.


