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{\bf Question}

Show that $G(x)=4x(1-x)$ has orbits of {\underline every} period.
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{\bf Answer}

$G^n$ has $2^n$ fixed points (from graph).  Of these, a maximum of
$2+4+\cdots+2^{n-1}=2^n-2$ can be fixed points of $G$ for $m&lt;n$,
so there are at least 2 points of period $n$.
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