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I need to prove that using the ordering of the Sharkocsky's, that period 4 implies period 2. Thus for a continuous function f from the unit interval to the unit interval itself, I need to prove that $f^4(x)=x$ implies $f^2(y)=y$. But I do not know how to do this. Can somebody help me with this proof. I also had to prove that period 2 implies period 1, but that can be done with the intermediate value theorm.

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Let $g=f^2$. Then $g$ has a periodic point of period $2$. Can you finish?

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    $\begingroup$ You need also to exclude that the 1-periodic point of $f^2$ that you get is not the 1-periodic point of $f$. $\endgroup$ – LutzL Oct 8 '18 at 17:53
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    $\begingroup$ That's part of finishing. $\endgroup$ – Julián Aguirre Oct 8 '18 at 17:55
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    $\begingroup$ @LutzL I see what you mean. It is not as simple as I thought. $\endgroup$ – Julián Aguirre Oct 8 '18 at 18:01

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