On the dichotomy of Perron numbers and beta-conjugates

Let β > 1 be an algebraic number. A general definition of a beta-conjugate of β is proposed with respect to the analytical function ${f_{beta}(z) =-1 + sum_{i geq 1} t_i z^i}$ associated with the Rényi β-expansion d _β (1) = 0.t ₁ t ₂ . . . of unity. From Szeg?’s Theorem, we study the dichotomy problem for f _β (z), in particular for β a Perron number: whether it is a rational fraction or admits the unit circle as natural boundary. The first case of dichotomy meets Boyd’s works. We introduce the study of the geometry of the beta-conjugates with respect to that of the Galois conjugates by means of the Erd?s–Turán approach and take examples of Pisot, Salem and Perron numbers which are Parry numbers to illustrate it. We discuss the possible existence of an infinite number of beta-conjugates and conjecture that all real algebraic numbers > 1, in particular Perron numbers, are in ${{rm C}_1 cup ,{rm C}_2 cup ,{rm C}_3}$ after the classification of Blanchard/Bertrand-Mathis.