A DOBROWOLSKI-TYPE INEQUALITY FOR THE POLES OF THE DYNAMICAL ZETA FUNTION OF THE BETA-SHIFT
Résumé
The poles of the dynamical zeta function of the β-shift in the open unit disk in C are investigated, when β > 1 is a real number close to 1 and tends to 1. We show that to β is associated an integer, called dynamical degree of β , and that two types of poles can be characterized as a function of dyg(β): the lenticular poles and the other poles in a narrow annular neighbourhood of the unit circle. From the lenticulus of lenticular poles we show that we can construct a lenticular measure associated to it. Its definition is mimicked on that of the Mahler measure, and both are related. We prove that this lenticular measure admits a lower bound, when dyg(β) is large enough, as a function of dyg(β), which is an analogue of the inequality of Dobrowolski relative to the Mahler measure of algebraic integers. This Dobrowolski-type inequality generalizes a previous one obtained by the author on the Mahler measure of the trinomials −1 + x + x^n for n large enough.
Origine : Fichiers produits par l'(les) auteur(s)