On the entropy of a family of random substitutions

The generalised random Fibonacci chain is a stochastic extension of the classical Fibonacci substitution and is defined as the rule that is mapping $${0\mapsto 1}$$ and $${1 \mapsto 1^i01^{m-i}}$$ with probability pi, where pi ≥ 0 with $${\sum_{i=0}^m p_i =1}$$ , and where the random rule is applied each time it acts on a 1. We show that the topological entropy of this object is given by the growth rate of the set of inflated generalised random Fibonacci words.