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

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