the decomposition of $\psi$. In the last section, we give some comments and state open questions.

Preliminaries
=============

An *alphabet* $\mathcal{A}$ is a finite set of symbols called *letters*. A *finite word* of length $n$ over $\mathcal {A}$ is a string $u=u_0u_1\cdots u_{n-1}$, where $u_i \in \mathcal{A}$ for all $i=0,1,\ldots, n-1$. The *length* of $u$ is denoted by $|u| = n$. By $|u|_a$ we denote the number of copies of the letter $a$ used in $u$, i.e. $|u|_a= \# \{i \in \mathbb{N} \colon  i< n, u_i = a\}$. The set of all finite words over $\mathcal{A}$ together with the operation of concatenation forms a monoid $\mathcal{A}^*$. Its neutral element is the *empty word* $\varepsilon$ and $\A^+ = \A^* \setminus  \left\{  \varepsilon \right\}$. On this monoid we work with two operations which preserve the length of words. The *mirror image* or *reversal* of a word $u=u_0u_1\cdots u_{n-1} \in \mathcal{A}^*$ is the word $\overline{u} = u_{n-1}u_{n-2}\cdots u_{1}u_0$. The *cyclic shift* of $u$ is the word $$\label{eq:def_of_cyc}
{\rm cyc}(u) = u_{1}u_{2}\cdots u_{n-1}u_0.$$

An *infinite word* over $\mathcal {A}$ is a sequence $\uu = u_0u_1u_2\cdots  = \left(u_i\right)_{i\in \mathbb{N}} \in \mathcal{A}^{\mathbb{N}}$ with $u_i \in \mathcal{A}$ for all $i \in \N = \left\{ 0,1,2, \ldots \right 