is in fact the image of the substitution applied to each letter of $w$, that is $\sigma(w_1w_2\cdots w_n) = \sigma(w_1) \sigma(w_2) \cdots \sigma(w_n)$, where $w_i$ are letters of the alphabet. As an example, the first few iterations on the letter $a$ using the Fibonacci substitution are given by $\sigma^1(a) = \sigma(a) = ab$, $\sigma^2(a) = \sigma(\sigma(a)) = \sigma(ab) = \sigma(a) \sigma(b) = aba$, $\sigma^3(a) = \sigma(\sigma^2(a)) = \sigma(aba) = \sigma(a)\sigma(b)\sigma(a) = abaab$, and so on (see [@Arnoux1991; @Vuillon2003]). Note that each word $\sigma^{n}(a)$ is the beginning of the next word $\sigma^{n+1}(a)$. Thus, as $n\to +\infty$, a non-periodic fixed point is constructed, viz. $$\sigma(F) = F =  
  abaababaabaababaababaabaababaabaababaababaabaababaababaabaababaaba\cdots.$$

On the other hand, a periodic word can be constructed by means of the substitution $\sigma(a)=ab$ and $\sigma(b)=ab$ by repetition of the pattern $ab.$. Indeed $\sigma^1(a)=\sigma(a)=ab, $$\sigma^2(a) = \sigma(\sigma(a)) = \sigma(ab) = \sigma(a)\sigma(b) = abab$, $\sigma^3(a) = \sigma(\sigma^2(a)) = \sigma(abab) = \sigma(a)\sigma(b)\sigma(a)\sigma(b) = abababab,$ and so on. The fixed point is periodic and given by $$\sigma(P) = P = 
  abababababababababababababababababababababababababababa\cdots.$$

In order to code periodic spatial patterns of the equation, one can exploit bi-dimensional words (see [@Frougny2005; @Berthe2000; @Lind1995]) and proper substitutions (see [@Arnoux1991; @Berthe2000; @Frougny2005]). For example, the bi-dimensional Thue–Morse substitution (see 