Rényi-Parry germs of curves and dynamical zeta functions associated with real algebraic numbers

Jean-Louis Verger-Gaugry (1)

1 IF - Institut Fourier

Let $\beta > 1$ be an algebraic number. The relations between the coefficient vector of its minimal polynomial and the digits of the Rényi $\beta$-expansion of unity are investigated in terms of the germ of curve associated with $\beta$, which is constructed from the Salem parametrization, and the Parry Upper function $f_{\beta}(z)$. If $\beta$ is a Parry number, the Parry Upper function $f_{\beta}(z)$ is simply related to the dynamical zeta function $\zeta_{\beta}(z)$ of the dynamical system $([0,1], T_{\beta})$ where $T_{\beta}$ is the $\beta$-transformation. Using the theory of Puiseux several results on the zeros of $f_{\beta}(z)$ and a classification of $\beta$s off Parry numbers are suggested.

Mots clés: Rényi-Parry numeration system; Pisot number; Salem number; Puiseux theory; germ of curve; Newton polygon; dynamical zeta function

Domaines: Théorie des nombres [math.NT]

Jean-Louis Verger-Gaugry. Rényi-Parry germs of curves and dynamical zeta functions associated with real algebraic numbers. RIMS Kôkyûroku Bessatsu, 2014, B46, pp.241-247. ⟨hal-00765631⟩