Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 4 / 7E / ii

Every nonempty relatively open interval contains a binary cylinder whose points share a finite prefix. Given nonempty open sets $U,V$, choose such a cylinder $C\subset U$ of length $N$. After $N$ binary shifts, $F^N(C)$ contains $[0,1)$ with the usual half-open cylinder convention. In particular, some point of $C$ maps into $V$. Thus $F$ is topologically transitive: $F^N(U)\cap V\ne \varnothing$. Equivalently, concatenate a prefix placing the point in $U$ with a tail placing its $N$th iterate in $V$.