Towards a statement of the S-adic conjecture through examples

TL;DR

This paper explores the S-adic conjecture by analyzing the factor complexity of S-adic sequences, providing examples that illustrate properties and challenge potential conditions for sub-linear complexity.

Contribution

It offers an overview of factor complexity in S-adic sequences and presents examples that test and refine the conjecture's proposed conditions.

Findings

Examples illustrating properties of S-adic sequences

Counter-examples to potential conditions for sub-linear complexity

Insights into the structure of S-adic sequences

Abstract

The $S$-adic conjecture claims that there exists a condition $C$ such that a sequence has a sub-linear complexity if and only if it is an $S$-adic sequence satisfying Condition $C$ for some finite set $S$ of morphisms. We present an overview of the factor complexity of $S$-adic sequences and we give some examples that either illustrate some interesting properties or that are counter-examples to what could be believed to be "a good Condition $C$".