Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 32B / a

A horseshoe consists of two disjoint intervals each mapped across their union, producing full two-symbol itinerary dynamics.

Glendinning chaos means there is an invariant set on which an iterate is semiconjugate to the full shift, giving sensitive dependence and periodic points of every period. For a 3-cycle ordered $x_1<x_2<x_3$, the intermediate-value theorem applied to the images of the intervening intervals constructs a horseshoe for an iterate; hence $F$ is chaotic.