[@Allouche2003]) is defined by $$\sigma(a) =
\begin{pmatrix}
  a & b \\ 
  b & a
\end{pmatrix}
\quad \mbox{and} \quad 
\sigma(b) = 
\begin{pmatrix}
  b & a \\ 
  a & b 
\end{pmatrix}.$$

One can just iterate a finite number of substitutions, viz. $\sigma^j(a)$ with $j$ finite, and obtain a finite world. For example, the first two iterations of the Thue-Morse substitution yield, respectively, the two words $$\sigma^1(a) = \sigma(a) = 
\begin{pmatrix}
  a & b \\ 
  b & a
\end{pmatrix}$$ and $$\sigma^2(a) = \sigma
\begin{pmatrix}
  a & b \\ 
  b & a
\end{pmatrix} = 
\begin{pmatrix}
  \sigma(a) & \sigma(b) \\ 
  \sigma(b) & \sigma(a)
\end{pmatrix} = 
\begin{pmatrix}
  a & b & b & a \\
  b & a & a & b \\
  b & a & a & b \\
  a & b & b & a
\end{pmatrix}$$

In the limit of infinite iterations, an infinite word is obtained in the form of the non-periodic fixed point $$\sigma(M) = M = 
\begin{pmatrix}
  a & b & b & a & b & a \cdots \\
  b & a & a & b & a & b \cdots \\
  b & a & a & b & a & b \cdots \\
  a & b & 