Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 32A / a

In Glendinning's definition, a continuous interval map $F$ is chaotic when $F^m$ has a horseshoe for some $m\ge 1$. Thus there are two disjoint subintervals whose images under $F^m$ each cover their union, producing full two-symbol itinerary dynamics.

A map is chaotic in Devaney's sense when it is topologically transitive, its periodic points are dense, and it has sensitive dependence on initial conditions. Explicitly, transitivity means that for every nonempty open $U,V$ some $n$ has $F^n(U)\cap V\ne \varnothing$, while sensitivity means that one $\delta >0$ works so that arbitrarily close initial points eventually separate by more than $\delta$.