On the Entropy of a Two Step Random Fibonacci Substitution

TL;DR

This paper studies a probabilistic variation of the Fibonacci substitution, deriving an exact formula for its topological entropy based on the growth rate of generated words, extending understanding of randomness in substitution systems.

Contribution

It introduces a new random Fibonacci substitution model and provides an exact calculation of its topological entropy, linking entropy to word growth rate.

Findings

Exact entropy formula derived for the random Fibonacci substitution.

Entropy depends on the probability parameter p.

Growth rate of random Fibonacci words determines topological entropy.

Abstract

We consider a random generalisation of the classical Fibonacci substitution. The substitution we consider is defined as the rule mapping $\mathtt{a}\mapsto \mathtt{baa}$ and $\mathtt{b} \mapsto \mathtt{ab}$ with probability $p$ and $\mathtt{b} \mapsto \mathtt{ba}$ with probability $1-p$ 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.