Dynamics of generic automorphisms of Stein manifolds with the density property

We study the dynamics of a generic automorphism f of a Stein manifold with the density property. Such manifolds include almost all linear algebraic groups. Even in the special case of, most of our results are new. We study the Julia set, non-wandering set and chain-recurrent set of f. We show that the closure of the set of saddle periodic points of f is the largest forward invariant set on which f is chaotic. This subset of the Julia set of f is also characterized as the closure of the set of transverse homoclinic points of f, and equals the Julia set if and only if a certain closing lemma holds. Among the other results in the paper is a generalization of Buzzard's holomorphic Kupka-Smale theorem to our setting.