On the beta expansion for Salem numbers of degree 6

Abstract:

For a given $\beta > 1$, 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 $\{T^{n}(1)\}_{n \ge 1}$ is finite, hence eventually periodic. It is known that all Pisot numbers are beta numbers, and it is conjectured that this is true for Salem numbers, but this is known only for Salem numbers of degree $4$. Here we consider some computational and heuristic evidence for the conjecture in the case of Salem numbers of degree $6$, by considering the set of $11836$ such numbers of trace at most $15$. Although the orbit is small for the majority of these numbers, there are some examples for which the orbit size is shown to exceed $10^{9}$ and for which the possibility remains that the orbit is infinite. There are also some very large orbits which have been shown to be finite: an example is given for which the preperiod length is $39420662$ and the period length is $93218808$. This is in contrast to Salem numbers of degree $4$ where the orbit size is bounded by $2\beta + 3$. An heuristic probabilistic model is proposed which explains the difference between the degree-$4$ and degree-$6$ cases. The model predicts that all Salem numbers of degree $4$ and $6$ should be beta numbers but that degree-$6$ Salem numbers can have orbits which are arbitrarily large relative to the size of $\beta$. Furthermore, the model predicts that a positive proportion of Salem numbers of any fixed degree $\ge 8$ will not be beta numbers. This latter prediction is not tested here.