Dynamical degrees of affine-triangular automorphisms of affine spaces

TL;DR

This paper investigates the dynamical degrees of affine automorphisms of affine spaces, providing a complete classification in dimension three and linking these degrees to weak Perron numbers, with bounds for quadratic integers.

Contribution

It characterizes all dynamical degrees in dimension three for affine-triangular automorphisms and connects these degrees to weak Perron numbers, establishing bounds for quadratic integers.

Findings

All dynamical degrees in dimension three are classified.

Every weak Perron number is realizable as a dynamical degree.

The minimal dimension for quadratic integers as dynamical degrees is 3 or 4.

Abstract

We study the possible dynamical degrees of automorphisms of the affine space $\mathbb{A}^n$. In dimension $n=3$, we determine all dynamical degrees arising from the composition of an affine automorphism with a triangular one. This generalises the easier case of shift-like automorphisms which can be studied in any dimension. We also prove that each weak Perron number is the dynamical degree of an affine-triangular automorphism of the affine space $\mathbb{A}^n$ for some $n$, and we give the best possible $n$ for quadratic integers, which is either $3$ or $4$.