Generating functions of substitutions

TL;DR

This paper establishes a connection between a substitution's aperiodicity and the transcendental nature of its associated generating functions, using recursive structures to analyze roots in specific cases like the Fibonacci substitution.

Contribution

It proves that a substitution is aperiodic if and only if some of its generating functions are transcendental, linking dynamical properties with transcendence.

Findings

A substitution is aperiodic iff some generating functions are transcendental.

Recursive structure of generating functions is used to analyze roots.

Specific analysis conducted for Fibonacci substitution.

Abstract

We prove that a substitution is aperiodic if and only if some of its associated generating functions are transcendental. These generating functions have a recursive structure arising from the substitution which we use to study their roots in the case of the Fibonacci substitution.