Remarks on the degree growth of birational transformations

Abstract

We look at sequences of positive integers that can be realized as degree sequences of iterates of rational dominant maps of smooth projective varieties over arbitrary fields. New constraints on the degree growth of endomorphisms of the affine space and examples of degree sequences are displayed. We also show that the set of all degree sequences of rational maps is countable; this generalizes a result of Bonifant and Fornaess.

Automorphisms of compact Kähler manifolds with slow dynamics

“…Except in dimension ≤ 2, we don't know whether s j (g) is an integer when λ j (g) = 1; it could a priori be the case that (g n ) * j grows like exp( √ n) or n √ 3 as n goes to +∞. We refer to [45] for this type of questions, and to [12,11] for the main properties of (g n ) * j .…”

Automorphisms and Dynamics: A List of Open Problems

“…One may also ask, as in Question 3.1, whether the equality λ 1 (f ) = 1 implies the existence of a non-trivial invariant fibration. While these questions are fully understood in dimension 2 (see [8,10,19,27]) almost nothing is known in higher dimension (see [44], as well as [18,36] for interesting examples).…”

A transcendental dynamical degree

“…For certain classes of maps, such as birational maps of P 2 [DF01] or polynomial maps of A 2 [FJ07, FJ11], we can achieve algebraic stability after birational conjugation; hence the dynamical degree is an algebraic integer in these cases. It has been shown, moreover, that there are only countably many different dynamical degrees among all rational maps, algebraically stable or not [BF00,Ure18].…”