Directed covering graph of an interval map

Given finitely many intervals

J_{1}, \dots, J_{r}

J_{i} \to J_{j}

whenever

F (J_{i}) \supseteq J_{j}

Periodic orbit from a closed interval-covering walk

Every closed walk

J_{0} ⟶ J_{1} ⟶ \dots ⟶ J_{n - 1} ⟶ J_{0}

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.

A

tr (A^{n})

counts its pointed closed walks of length

n

p

, subtracting the

tr (A)

constant walks and identifying the

p

cyclic choices of starting point gives

\frac{tr ( A ^{p} ) - tr ( A )}{p}

primitive closed itineraries of length

p

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