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

Choose binary expansions $x=0.b_1{b}_2\dots$ that do not end in an infinite string of ones. The doubling map deletes the first digit: $F(x)=0.b_2{b}_3\dots$. Given $x$ and any neighbourhood of it, choose $N$ sufficiently large that its binary cylinder of length $N$ is small enough, and choose a point $y$ sharing that prefix but with a prescribed tail. After $N$ shifts, the tail of $y$ may be chosen arbitrarily in $[0,1)$. Choose it close to zero or close to one, whichever is at distance more than $1/3$ from $F^N(x)$. Then $|x-y|\le 2^{-N}$ but $|F^N(x)-F^N(y)|>1/3$. Hence the map has sensitive dependence on initial conditions, with a uniform separation constant. At dyadic endpoints the cylinder sharing the chosen terminating expansion still contains points arbitrarily close to the endpoint.