Construct a continuous function $F:[0,1] \to [0,1]$ that has a point with period 2015.
I think I should do it with Sharkovskii's theorem, but then? Where can I start with?
Or try to find a point of period 3 and by LiYorke theorem?
2026-03-27 03:00:10.1774580410
Construct a continuous function $F:[0,1]\rightarrow[0,1]$ that has a point with period 2015
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How about something as simple as $$F(x)=\begin{cases}x-\frac{1}{2014}& \text{for }x>\frac{1}{2014}\\ 1-2014x & \text{for } x\leq \frac{1}{2014} \end{cases}$$
Then $f^{2015}(1)=1$, and $f^n(1)\neq 1$ for $n<2015.$
Addendum
Of course this works for every natural period. Here's what the cobweb plot looks like when when the period is $15$.