Salem/Pisot Numbers in the Weyl Spectrum

TL;DR

The article develops a rigorous bridge between birational dynamics on $\mathbf{P}^2(\mathbb{C})$ and Coxeter-theoretic Weyl groups by introducing orbit data that encodes exceptional-curves behavior under iteration. It proves that the Weyl spectrum $\Lambda_W$, consisting of dynamical degrees, is determined recursively via Weyl degree and orbit data, and it identifies all Salem and Pisot numbers within this spectrum through finite, computable sets $\mathcal{M}_d$. It also shows that every accumulation point of Pisot numbers not exceeding $2$ lies in the Weyl spectrum and provides explicit constructions and limit processes for realizing small Pisot numbers as dynamical degrees. The results unify the dynamical spectrum with the Weyl spectrum, give explicit characteristic polynomials $\chi_{M,\bar{n}}(t)$ that govern spectral radii, and supply concrete examples and methods to realize Pisot numbers as dynamical degrees of birational maps on $\mathbf{P}^2(\mathbb{C})$, enhancing understanding of which algebraic integers occur as growth rates in complex dynamics.

Abstract

n 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.

Figures (1)

• Figure 1: Coxeter Graph $E_n$

Theorems & Definitions (70)

• Remark 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Proposition 2.1
• proof
• Definition 2.2
• Proposition 2.3
• proof
• Theorem 2.4