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