ATTRACTING BASINS OF VOLUME PRESERVING AUTOMORPHISMS OF Ck

Abstract. We study topological properties of attracting sets for automor-phisms of Ck. Our main result is that a generic volume preserving automor-phism has a hyperbolic fixed point with a dense stable manifold. We prove the same result for volume preserving maps tangent to the identity. On the other hand, we show that an attracting set can only contain a neighborhood of the fixed point if it is an attracting fixed point. We will see that the latter does not hold in the non-autonomous setting. 1. introduction Let f be an automorphism of Ck with a fixed point at the origin. Even if the origin is not an attracting fixed point, there can still be points whose orbits converge to the origin. In this paper we will study how large such an attracting set can be. We will make this more precise later. The behavior of an attracting set varies greatly depending on the eigenvalues of df(0). If all eigenvalues have modulus strictly smaller than 1 then we say that f has an attracting fixed point. This is the easiest situation, the attracting se