On the Entropy of a Family of Random Substitutions

TL;DR

This paper investigates the entropy of a stochastic extension of the Fibonacci substitution, showing that its topological entropy equals the growth rate of a specific set of generated words.

Contribution

It provides a formula linking the topological entropy of the generalized random Fibonacci chain to the growth rate of associated word sets.

Findings

Topological entropy is characterized by the growth rate of inflated words.

The entropy formula applies to a family of random substitutions.

The results extend understanding of entropy in stochastic substitution systems.

Abstract

The generalised random Fibonacci chain is a stochastic extension of the classical Fibonacci substitution and is defined as the rule mapping $0\mapsto 1$ and $1 \mapsto 1^i01^{m-i}$ with probability $p_i$, where $p_i\geq 0$ 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.