Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 32A / a

Concatenate every finite binary word, ordered first by length and then lexicographically:Every finite binary block occurs in this expansion. Since iterating $F$ shifts digits to the left, its orbit enters every binary cylinder and is therefore dense. The digit string is not eventually periodic, so it gives a concrete nonperiodic chaotic orbit.

A fixed point satisfiesso $x$ is an integer. In $[0,1)$ the only possibility isEquivalently, a binary sequence fixed by the left shift is constant; $0.000\dots {}_2=0$, while $0.111\dots {}_2=1$ represents the excluded endpoint, which is the same circle point.

Points whose period divides two solveso, after removing the fixed point zero, the unique two-cycle is

Points whose period divides four are $j/15$ for $j=0,\dots ,14$. Removing the three points fixed by $F^2$, namely $0,5/15,10/15$, leaves twelve points of exact period four. They form the three orbitsandStarting at another point on one displayed orbit gives the corresponding cyclic rotation.