Automorphisms of the affine 3-space of degree 3

Abstract: 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 $\mathbb{A}^2$.