= Periodic orbit from a closed interval-covering walk
Every closed walk
$$
J_0\longrightarrow J_1\longrightarrow\cdots\longrightarrow J_{n-1}\longrightarrow J_0
$$
in a <directed covering graph of an interval map> has a point $x\in J_0$ with $F^j(x)\in J_j$ and $F^n(x)=x$. This follows by successively pulling $J_0$ back through the covering relations and applying the <fixed-point property of a closed interval>. If the itinerary has least period $n$ and avoids shared endpoints, $x$ lies on an $n$-cycle.
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