# Inflation word entropy for semi-compatible random substitutions

## TL;DR

This paper introduces inflation word entropy for semi-compatible random substitutions, proving its equality with topological entropy and providing methods for its approximation and explicit calculation.

## Contribution

It establishes a unified framework linking inflation word entropy with topological entropy, including proofs and approximation techniques.

## Key findings

- Inflation word entropy equals topological entropy.
- Series of bounds can approximate topological entropy.
- Analytic expressions are obtainable in many cases.

## Abstract

We introduce the concept of inflation word entropy for random substitutions with a constant and primitive substitution matrix. Previous calculations of the topological entropy of such systems implicitly used this concept and established equality of topological entropy and inflation word entropy, relying on ad hoc methods. We present a unified scheme, proving that inflation word entropy and topological entropy in fact coincide. The topological entropy is approximated by a converging series of upper and lower bounds which, in many cases, lead to an analytic expression.