On the Lyndon Dynamical Systems

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 $\beta >1$.