On the Entropy of a Two Step Random Fibonacci Substitution

We consider a random generalisation of the classical Fibonacci substitution. The substitution we consider is defined as the rule mapping $$\mathtt{a}$\mapsto $\mathtt{b}\mathtt{a}\mathtt{a}$$and$$\mathtt{b}$ \mapsto $\mathtt{a}\mathtt{b}$$withprobability$p$and$$\mathtt{b}$ \mapsto $\mathtt{b}\mathtt{a}$$withprobability$1-p$for$0<p<1$ and where the random rule is applied each time it acts on a $\mathtt{b}$. We show that the topological entropy of this object is given by the growth rate of the set of inflated random Fibonacci words, and we exactly calculate its value.