On the Lyndon dynamical system

Given a totally finite ordered alphabet AA, endowing the set of words over AA with the alternating lexicographic order (see [6]), 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 (see [11]), the conditions for which a word is the (−β)(−β)-expansion of −ββ+1 for some β>1β>1.