  We study the dynamics of a generic automorphism $f$ of a Stein manifold with
the density property. Such manifolds include all linear algebraic groups. Even
in the special case of $\mathbb C^n$, $n\geq 2$, 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 characterised 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 generalisation
of Buzzard's holomorphic Kupka-Smale theorem to our setting.
