On finite $\beta$-expansions for the set of natural numbers

We present a study of the problem of finiteness of the $\beta$-expansions for the set of natural numbers, condition $F_1$ in brief, for three families of Pisot numbers for which the $\beta$-expansion of 1 is not a non-decreasing sequence. We show a class of simple $\beta$-numbers which display a puzzling behaviour, in the sense that an infinite subset of such $\beta$ satisfy $F_1$, whereas a complementary infinite subset does not. This puzzle is organized by the residue class modulo 3 of an integer coefficient of the main family of polynomials discussed. We prove that, when $F_1$ does not hold, the complement of $\mathrm{F}\mathrm{i}\mathrm{n}(\beta )$ is infinite. Finally, we give a concise sufficient condition for $F_1$, which is the finitude of the $\beta$-expansion of $(\beta -\lfloor \beta \rfloor {)}^2$.