Beyond substitutive dynamical systems: S-adic expansions

TL;DR

This paper explores S-adic expansions of infinite words, connecting combinatorics, ergodic theory, and Diophantine approximation, and relating them to continued fractions and classical sequences like Sturmian sequences.

Contribution

It provides a comprehensive analysis of S-adic words from multiple mathematical perspectives, highlighting their relation to continued fractions and classical sequences.

Findings

S-adic expansions generalize continued fractions for words.

Connections between Sturmian sequences and regular continued fractions are elucidated.

Multiple perspectives deepen understanding of S-adic structures.

Abstract

An S-adic expansion of an infinite word is a way of writing it as the limit of an infinite product of substitutions (i.e., morphisms of a free monoid). Such a description is related to continued fraction expansions of numbers and vectors. A fundamental example of this relation is between Sturmian sequences and regular continued fractions. We study S-adic words from different perspectives, namely word combinatorics, ergodic theory, and Diophantine approximation, by stressing the parallel with continued fraction expansions.