On the dichotomy of Perron numbers and higher-order Parry numbers

Jean-Louis Verger-Gaugry Université Joseph Fourier, Grenoble, France

Let $\beta >1$ be an algebraic number. From Szego's Theorem (1922), we know the dichotomy problem for $\beta$ : whether the analytical function $f_{\beta}(z) = -1 + \sum_{i=1}^{\infty} t_i z^i$ associated with the Rényi $\beta$-expansion $d_{\beta}(1) = 0 . t_1 t_2 t_3 \ldots$ of unity is a rational fraction or admits the unit circle as natural boundary.
The theorem we present shows that if $\beta$ is such that $f_{\beta}(z)$ represents an algebraic function, of degree $n\geq 2$, then any string of any letter in $d_{\beta}(1)$ is bounded, in particular gaps (strings of zeros), and $d_{\beta}(1)$ is aperiodic. In this case $\beta$ enters the second case of the dichotomy. This theorem concerns the class C$_3$, or else the class Q $_{0}^{(2)}$. We discuss the case of Perron numbers which arise from dynamical substitution systems over finite alphabets. The case $n=1$ corresponds to Parry numbers, i.e. Parry numbers of degree one. When $n\geq 2$, we call $\beta$ a Parry number of degree $n$. We give an upper bound of the maximal length of the strings $\gamma \gamma \ldots \gamma$ of a letter $\gamma$ in $d_{\beta}(1)$, valid for any letter $\gamma$ of the alphabet, as a function of $n$ and of the polynomials entering the irreducible form of the equation of the algebraic function.