2 7 Fe b 20 09 Continuous Families of Rational Surface Automorphisms with Positive Entropy Eric Bedford

§0. Introduction. Cantat [C1] has shown that if a compact projective surface carries an automorphism of positive entropy, then it has a minimal model which is either a torus, K3, or rational (or a quotient of one of these). It has seemed that rational surfaces which carry automorphisms of positive entropy are relatively rare. Indeed, the first infinite family of such rational surfaces was found only recently (see [BK1,2] and [M]). Here we will show, on the contrary, that positive entropy rational surface automorphisms are more “abundant” than the torus and K3 cases, in the sense that they are contained in families of arbitrarily high dimension. We define our automorphisms in terms of birational models. We say that a birational map f of P is an automorphism if there are a rational surface X = Xf , an iterated blowup map π : X → P, and an automorphism fX of X such that π◦fX = f ◦π. We will consider birational maps of the form