On beta expansions for Pisot numbers

Abstract:

Given a number $\beta > 1$, the beta-transformation $T =T_{\beta }$ is defined for $x \in [0,1]$ by $Tx := \beta x$ (mod 1). The number $\beta$ is said to be a beta-number if the orbit $\{T^{n}(1)\}$ is finite, hence eventually periodic. In this case $\beta$ is the root of a monic polynomial $R(x)$ with integer coefficients called the characteristic polynomial of $\beta$. If $P(x)$ is the minimal polynomial of $\beta$, then $R(x) = P(x)Q(x)$ for some polynomial $Q(x)$. It is the factor $Q(x)$ which concerns us here in case $\beta$ is a Pisot number. It is known that all Pisot numbers are beta-numbers, and it has often been asked whether $Q(x)$ must be cyclotomic in this case, particularly if $1 < \beta < 2$. We answer this question in the negative by an examination of the regular Pisot numbers associated with the smallest 8 limit points of the Pisot numbers, by an exhaustive enumeration of the irregular Pisot numbers in $[1,1.9324]\cup [1.9333,1.96]$ (an infinite set), by a search up to degree $50$ in $[1.9,2]$, to degree $60$ in $[1.96,2]$, and to degree $20$ in $[2,2.2]$. We find the smallest counterexample, the counterexample of smallest degree, examples where $Q(x)$ is nonreciprocal, and examples where $Q(x)$ is reciprocal but noncyclotomic. We produce infinite sequences of these two types which converge to $2$ from above, and infinite sequences of $\beta$ with $Q(x)$ nonreciprocal which converge to $2$ from below and to the $6$th smallest limit point of the Pisot numbers from both sides. We conjecture that these are the only limit points of such numbers in $[1,2]$. The Pisot numbers for which $Q(x)$ is cyclotomic are related to an interesting closed set of numbers $\mathcal {F}$ introduced by Flatto, Lagarias and Poonen in connection with the zeta function of $T$. Our examples show that the set $S$ of Pisot numbers is not a subset of $\mathcal {F}$.