On the Lyndon Dynamical Systems

TL;DR

This paper introduces a new class of Lyndon words based on an alternating lexicographic order, explores their associated dynamical systems called Lyndon systems, and relates these to properties of negative base beta-shifts, providing new conditions for specific expansions.

Contribution

It defines a novel class of Lyndon words with an associated dynamical system and establishes independent conditions for negative base expansions, expanding understanding of symbolic dynamics.

Findings

Characterization of Lyndon systems with alternating lex order

Relation between Lyndon systems and negative base beta-shifts

New conditions for the $(-eta)$-expansion of specific numbers

Abstract

Given a totally finite ordered alphabet $ A $, endowing the set of words over $ A $ with the alternating lexicographic order, we define a new class of Lyndon words. We study the fundamental properties of the associated symbolic dynamical systems called Lyndon system. We derive some fundamental properties of the beta-shift with negative base by relating it with the Lyndon system. We find, independently of W. Steiner's method, the conditions for which a word is the $(-\beta)$-expansion of $-\frac{\beta}{\beta + 1} $ for some $ \beta> 1$.