An Analogue of the Thue-Morse Sequence

Abstract

We consider the finite binary words $Z(n)$, $n \in {\Bbb N}$, defined by the following self-similar process: $Z(0):=0$, $Z(1):=01$, and $Z(n+1):=Z(n)\cdot\overline{Z(n-1)}$, where the dot $\cdot$ denotes word concatenation, and $\overline{w}$ the word obtained from $w$ by exchanging the zeros and the ones. Denote by $Z(\infty)=01110100 \dots$ the limiting word of this process, and by $z(n)$ the $n$'th bit of this word. This sequence $z$ is an analogue of the Thue-Morse sequence. We show that a theorem of Bacher and Chapman relating the latter to a "Sierpiński matrix" has a natural analogue involving $z$. The semi-infinite self-similar matrix which plays the role of the Sierpiński matrix here is the zeta matrix of the poset of finite subsets of ${\Bbb N}$ without two consecutive elements, ordered by inclusion. We observe that this zeta matrix is nothing but the exponential of the incidence matrix of the Hasse diagram of this poset. We prove that the corresponding Möbius matrix has a simple expression in terms of the zeta matrix and the sequence $z$.