Automorphisms of the affine 3-space of degree 3

Abstract

In this article we give two explicit families of automorphisms of degree ≤ 3 of the affine 3-space A 3 such that each automorphism of degree ≤ 3 of 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 ≤ 3 of 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 ≥ 1 all morphisms A 3 → A n of degree ≤ 3 with the property that the preimage of every affine hyperplane in A n is isomorphic to A 2 .

A central limit theorem for the degree of a random product of Cremona transformations

“…As a result, the growth of the sequence deg(f n ) and the dynamical degree of a given rational map f has been the subject of much research, and is known only in certain cases: for endomorphisms of a projective variety, monomial maps [Lin12,FW12], birational surfaces maps [DF01,BC16], polynomial automorphisms and endomorphisms of the affine plane [FM89, Fur99, FJ04, FJ07, FJ11], meromorphic surface maps (under certain assumptions) [BFJ08], birational transformations of hyperkähler manifolds [LB19], certain automorphisms of the affine 3-space [BvS19a,BvS19b] and certain rational maps associated to matrix inversions [AAdBM99, AdMV06, AAdB + 99, BK08, BT10]. Starting from dimension 3, the degree sequences are partially known for birational transformations with very slow degree growth [CX18] or for specific examples [D 18], for a specific group of automorphisms on SL 2 (C) [Dan18], while a lower bound on unbounded degree sequences was recently obtained for a large class of birational transformations in [LU20].…”

Spectral interpretations of dynamical degrees and applications

“…These conjectures hold true in dimension 2 by Friedland-Milnor [FM89] and Favre-Jonsson [FJ11]. Computations for specific families of polynomial maps have been made in dimension 3, including quadratic automorphisms (Maegawa [Mae01]); cubic and triangular automorphisms (Blanc and van Santen [BvS19a,BvS19b]); and shift-like automorphisms (Jonsson, unpublished, see [BvS19b, §4.2]). These results support both conjectures.…”