On the Entropy of a Two Step Random Fibonacci Substitution

We consider a random generalization of the classical Fibonacci substitution. The substitution we consider is defined as the rule mapping, a 7→ baa and b 7→ ab, with probability p, and b 7→ ba, with probability 1 − p for 0 < p < 1, and where the random rule is applied each time it acts on a 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.