Nondensity of stability for polynomial automorphisms of C 2 Gregery T. Buzzard Abstract In the space of polynomial automorphisms of C 2 , the set of structurally stable maps is not dense. To obtain this result we develop some of the theory of moduli of stability for holomorphic maps. Also, the set of hyperbolic maps is not dense in the set of polynomial automorphisms; however, given any polynomial automorphism, F , of C 2 , there is a polynomial automorphism, G, of the same degree as and arbitrarily near F such that each periodic point of G is hyperbolic. 1 Introduction In this paper we consider questions of dynamical stability; i.e., when do small changes in a map lead to small changes in the dynamics? More precisely, let G be a topological space of diffeomorphisms of a manifold M . A map F ∈G is structurally stable if there exists a neighborhood U of F such that for each G ∈U , F and G are conjugate. In the study of dynamics of one complex variable, the question of the density of stability was resolved in [MSS] and [MS], where it was shown that in a family of rational maps of the Riemann sphere which depends holomorphically on a parameter, the set of structurally stable maps is dense. In contrast, for general C k diffeomorphisms of a compact surface, the set of structurally stable maps is not dense in most families; e.g., [I, Ch. 7, Sec. IV]. In this paper, we show that the situation for polynomial automorphisms of C 2 of fixed degree is analogous to that for C k diffeomorphisms of a compact surface: structural stabil- ity is not dense. In doing so, we develop some of the theory of moduli of stability along the lines of [NPT]. In this particular case, there are numerical invariants associated with homoclinic and heteroclinic tangencies which can prevent conjugation between maps having different values for these invariants. We also show that hyperbolic maps are not dense in the space of polynomial automorphisms of sufficiently high degree, but that given a polynomial automorphism of degree d having nontrivial dynamics, there is a nearby automorphism of the same degree such that all of its periodic points are hyperbolic. This latter result is one half of the Kupka-Smale theorem in the setting of polynomial automorphisms. Friedland and Milnor initiated the study of the dynamics of polynomial automorphisms of C 2 in [FM]. There they distinguish between elementary automorphisms, which are polyno- mially conjugate to an automorphism of the form (x,y ) → (ax + p(y ),cy + d)(p polynomial, a,c = 0) and which have simple dynamics, and the remaining automorphisms, which are termed nonelementary and which do not have simple dynamics. Bedford and Smillie [BS2] introduced the notion of the dynamical degree of a polynomial automorphism of C 2 . Letting deg F denote the maximum of the degrees of the polynomial map F , the dynamical degree is defined as d = d(F ) = lim n→∞ (deg F n ) 1/n , 1