Salem/Pisot numbers in the Weyl spectrum

Abstract

In this article, we define orbit data for birational maps of ${\mathbf{P}}^2(\mathbb{C})$ and show that this data uniquely determines the dynamical degree by providing minimal polynomials for dynamical degrees in terms of orbit data. Leveraging this relationship, we recursively identify all Salem and Pisot numbers that appear in the Weyl spectrum of the union of the Coxeter groups $W_n$ associated with $E_n$ and the set of all dynamical degrees of birational maps of ${\mathbf{P}}^2(\mathbb{C})$. Furthermore, we demonstrate that all accumulation points of Pisot numbers less than or equal to $2$ are present in the Weyl spectrum.