Horseshoe for an interval map

An interval map $F:I\to I$ has a horseshoe when there are an open interval $J\subseteq I$ and disjoint open intervals $K_0,K_1\subseteq J$ such thatIterating inverse branches produces full two-symbol itinerary dynamics.

If a directed covering graph of an interval map has two distinct closed walks of the same length $n$, based at the same interval and with different first edges, successive use of the interval covering relation produces two subintervals with disjoint interiors that $F^n$ maps across the base interval. Hence $F^n$ has a horseshoe for an interval map.