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Periodic orbit from a closed interval-covering walk

Every closed walk

J_{0} ⟶ J_{1} ⟶ \dots ⟶ J_{n - 1} ⟶ 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.

Table of contents 101 1
- Counting cycles in an interval covering graph Periodic orbit from a closed interval-covering walk 49

 Ancestors (8)