Canonical height functions for affine plane automorphisms

Abstract

Abstract. Let f : A 2 → A 2 be a polynomial automorphism of dynamical degree δ ≥ 2 over a number field K. (This is equivalent to say that f is a polynomial automorphism that is not triangularizable.) Then we construct canonical height functions defined on A 2 (K) associated with f . These functions satisfy the Northcott finiteness property, and an Kvalued point on A 2 (K) is f -periodic if and only if its height is zero. As an application of canonical height functions, we give an estimate on the number of points with bounded height in an infinite f -orbit.

Canonical heights for Hénon maps

“…In particular, such a map has only finitely many Krational periodic points. This result was subsequently extended and generalized by Denis [7], Kawaguchi [12], and Marcello [15]. The purpose of this note is to explore further the canonical heights associated to Hénon maps, that is, maps of the form ϕ(x, y) = (ay, x + f (y)), where f is a polynomial of degree at least 2, defined over a number field or function field.…”

Projective Surface Automorphisms of Positive Topological Entropy from an Arithmetic Viewpoint

“…Then the modified assertion holds true with Z 1 = Z 2 = Z 3 = H ∞ , where H ∞ is the hyperplane at infinity, i.e., H ∞ = P 2 \ A 2 . For details, see [13] and the references therein. We warn, however, that there are (X, f ) for which the modified assertion does not hold true.…”

Height Bounds and Preperiodic Points for Families of Jointly Regular Affine Maps

“…Kawaguchi [8] returns to the idea of resolving rational maps via blowups and gives two proofs of Conjecture 3 for all regular affine automorphisms in dimension 2. The first uses an ingeneious intersection theory argument and the second uses an explicit resolution of Hénon maps by Hubbard, Papadopol and Veselov [7].…”

“…Kawaguchi [8] has suggested an improvement in the lower bound in Corollary 2 and has given a proof in dimension 2. Although the improvement may appear to be minor, it has important consequences for the construction of canonical height functions.…”

“…The geometry and arithmetic of regular affine automorphisms has been the object of considerable study, see for example [8,13,16,19]. More generally, we define a collection of morphisms (polynomial maps)…”