Automorphisms of the affine 3-space of degree 3

In this article we give two explicit families of automorphisms of degree \(\leq 3\) of the affine \(3\)-space \(\mathbb{A}^3\) such that each automorphism of degree \(\leq 3\) of \(\mathbb{A}^3\) 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 \(\leq 3\) of \(\mathbb{A}^3\); this is a set of \(3\) integers and \(9\) quadratic integers. Moreover, we also describe up to compositions with affine automorphisms for \(n\geq 1\) all morphisms \(\mathbb{A}^3 \to \mathbb{A}^n\) of degree \(\leq 3\) with the property that the preimage of every affine hyperplane in \(\mathbb{A}^n\) is isomorphic to \(\mathbb{A}^2\).