MSS sequences, colorings of necklaces, and periodic points of \(f(z)=z^ 2-2\) (Q1100468)

For a map f of (0,1) into itself and a value \(\lambda\in (0,1)\), we may form a finite or possibly infinite sequence of R's and L's, \(\{b_ i\}\), by considering the iterates of the map \(\lambda\) f at 1/2. For \(i\geq 1\), set: (1) \(b_ i=R\), if \((\lambda f)^ i(1/2)>1/2;\) (2) \(b_ i=L\), if \((\lambda f)^ i(1/2)<1/2;\) (3) \(b_ i=C\), if \((\lambda f)^ i(1/2)=1/2.\) If \(b_ i=C\) for some i, then the sequence stops. Finite sequences of R's and L's obtained in this manner are called MSS sequences. The author proves that the number of MSS sequences of length n, for all \(n\in N\), is the same as the number of distinct negative orbits of order n using the function \(f(z)=z^ 2-2.\) In the last section of the paper, the algorithm is presented, which for each \(n\in N\) produces a bijection between the set of all MSS sequences of length n and the set of all primitive colorings of a necklace consisting of n beads.