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

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.