Canonical Height Functions on the Affine Plane Associated with Polynomial Automorphisms

Let f : A → A 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(K) associated with f . These functions satisfy the Northcott finiteness property, and an Kvalued point on A(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. Introduction and the statement of the main results One of the basic tools in Diophantine geometry is the theory of height functions. On Abelian varieties defined over a number field, Néron and Tate developed the theory of canonical height functions that behave well relative to the [n]-th power map (cf. [9, Chap. 5]). On certain K3 surfaces with two involutions, Silverman [14] developed the theory of canonical height functions that behave well relative to the two involutions. For the theory of canonical height functions on some other projective varieties, see for example [1], [16], [7]. In this paper, we show the existence of canonical height functions on the affine plane associated with polynomial automorphisms of dynamical degree ≥ 2. Consider a polynomial automorphism f : A → A given by f ( x y ) = ( p(x, y) q(x, y) ) , where p(x, y) and q(x, y) are polynomials in two variables. The degree d of f is defined by d := max{deg p, deg q}. The dynamical degree δ of f is defined by δ := lim n→+∞ (deg f) 1 n , which is an integer with 1 ≤ δ ≤ d. We let d ≥ 2. Polynomial automorphisms with δ = d are exactly regular polynomial automorphisms. Here a polynomial automorphism f : A → A is said to be regular if the unique point of indeterminacy of f is different from the unique point of indeterminacy of f−1, where the birational map f : P 99K P (resp. f−1 : P 99K P) is the extension of f (resp. f). In the moduli of polynomial automorphisms of degree d, regular polynomial automorphisms constitute general members, including Hénon maps. The other extreme is polynomial automorphisms of dynamical degree δ = 1, and they are exactly triangularizable automorphisms. Here a polynomial automorphism f : A → A is 1991 Mathematics Subject Classification. 11G50, 32H50.