= Period-three-free five-cycle interval covering pattern
For cycle points ordered $x_3<x_1<x_0<x_2<x_4$, successive gap vertices have arrows $0\to3$, $1\to1,2$, $2\to0$, $3\to0,1$. Their transition-matrix traces through length four are $1,3,1,7$, forcing periods one, two and four but not three. The affine interpolation through ordered values $(4,3,1,0,2)$ has no three-cycle: its only length-three closed interval itinerary is $111$, on which the map is $5-2x$ and its third iterate has only its fixed point. Two distinct length-four return words at vertex one give a horseshoe for the fourth iterate.
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