= Monotone five-cycle interval covering pattern
A five-cycle whose points are in orbit order along the real line gives four interval vertices with arrows $0\to1\to2\to3$ and $3\to0,1,2,3$. The transition-matrix traces for lengths one through four are $1,3,7,15$. Primitive closed words force at least one fixed point, one two-cycle, two three-cycles and three four-cycles. The second iterate has a two-branch <horseshoe for an interval map> on vertices two and three. The map itself need not have a one-step horseshoe: the unimodal affine example $F(x)=x+1$ for $x\le3$, $F(x)=16-4x$ for $x\ge3$ has precisely this five-cycle and has no two disjoint self-covering branches.
Back to article page
