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

For the Fibonacci transition matrix above,where $L_n$ is the $n$th Lucas number. It counts closed symbolic itineraries of length $n$. Since $7$ is prime, the itineraries fixed by the seventh shift have least period either $1$ or $7$. Hence the number of cyclic classes of primitive length-seven itineraries isDistinct primitive cyclic itineraries give distinct cycles, so $F$ has at least four distinct $7$-cycles. This is the fibonacci transition-graph cycle count.