Automorphisms of the affine 3-space of degree 3
J\'er\'emy Blanc, Immanuel van Santen

TL;DR
This paper classifies all automorphisms of degree up to 3 of the affine 3-space, showing they are all tame and providing a complete list of their dynamical degrees, with implications for morphisms with hyperplane preimages.
Contribution
It explicitly characterizes all degree ≤ 3 automorphisms of affine 3-space as members of two families, proving they are all tame and enumerating their dynamical degrees.
Findings
All automorphisms of degree ≤ 3 are tame.
Complete list of dynamical degrees includes 3 integers and 9 quadratic integers.
Classification of morphisms with hyperplane preimages is provided.
Abstract
In this article we give two explicit families of automorphisms of degree of the affine -space such that each automorphism of degree of is a member of one of these families up to composition of affine automorphisms at the source and target; this shows in particular that all of them are tame. As an application, we give the list of all dynamical degrees of automorphisms of degree of ; this is a set of integers and quadratic integers. Moreover, we also describe up to compositions with affine automorphisms for all morphisms of degree with the property that the preimage of every affine hyperplane in is isomorphic to .
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Algebraic Geometry and Number Theory · Geometric and Algebraic Topology
