Dynamics of Rational Surface Automorphisms: Rotation Domains

§0. Introduction. Let X denote a compact complex surface, and let f be a (biholomorphic) automorphism of X . The regular part of the dynamics of f occurs on the Fatou set F(f) ⊂ X , where the forward iterates are equicontinuous. As in [BS, U], we call a Fatou component U ⊂ F(f) a rotation domain of rank d if f |U generates a (real torus) T-action on U . In dimension 1, rotation domains correspond to Siegel disks or Herman rings, which have a (circle) Taction. Here we consider surface automorphisms with the property that the induced map f on H2(X ) has an eigenvalue greater than one. This is equivalent to the condition that f have positive entropy. Let us consider generally the possibilities of Fatou sets for surface automorphisms. If X is a complex 2-torus, then an automorphism with positive entropy is essentially an element of GL(2,Z). Positive entropy implies that the eigenvalues are |λ1| < 1 < |λ2|, and in this case the Fatou set is empty. A second possibility is given by K3 surfaces (or certain quotients of them). Since there is an invariant volume form, the only possible Fatou components are rotation domains. McMullen [M1] has shown the existence of non-algebraic K3 surfaces with rotation domains of rank 2 (see also [O]). By Cantat [C], the only other possibilities for compact surfaces with automorphisms of positive entropy are rational surfaces. In fact, by [BK2], the rational case is the most “frequent.” By definition, a rational surface is birationally (or bimeromorphically) equivalent to P, and by a result of Nagata, we may assume that it is obtained by iterated blowups of P. Rotation domains of rank 1 and 2 have been shown to occur for rational surface automorphisms (see [M2] and [BK1]). Other maps in this family of rational surface automorphisms were found to have attracting and/or repelling basins (see [M2] and [BK1]). In this paper we show that positive entropy automorphisms can have large rotation domains. To describe this, let Σ0 ⊂ P be the line at infinity. We will construct a complex manifolds π : X → P by performing iterated blowups to level 3 over points {p0, . . . , pn−1} ⊂ Σ0. We let F1 s denote the fiber obtained by blowing up ps, and at level 2 we denote by F2 s the fiber obtained by blowing up a point qs ∈ F1 s . We construct a pair (H,X ) with a rotation domain which corresponds to Figure 1: Theorem A. There is a rational surface X with an automorphism H which has positive entropy, and a rotation domain U ⊃ Σ0 ∪ F1 0 ∪ · · · ∪ F1 n−1. U is the union of invariant (Siegel) disks on each of which H acts as an irrational rotation.