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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 
${\mathbb {C}}^n$, 
$n\geq 2$, 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 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.
Let 
$Y$ be an infinite covering space of a projective manifold 
$M$ in 
${{\mathbb{P}}^{N}}$ of dimension 
$n\ge 2$ . Let 
$C$ be the intersection with $M$ of at most 
$n-1$ generic hypersurfaces of degree 
$d$ in ${{\mathbb{P}}^{N}}$ . The preimage 
$X$ of $C$ in $Y$ is a connected submanifold. Let 
$\phi$ be the smoothed distance from a fixed point in $Y$ in a metric pulled up from $M$ . Let 
${{\mathcal{O}}_{\phi }}(X)$ be the Hilbert space of holomorphic functions 
$f$ on $X$ such that 
${{f}^{2}}{{e}^{-\phi }}$ is integrable on $X$ , and define ${{\mathcal{O}}_{\phi }}(X)$ similarly. Our main result is that (under more general hypotheses than described here) the restriction 
${{\mathcal{O}}_{\phi }}(Y)\to {{\mathcal{O}}_{\phi }}(X)$ is an isomorphism for $d$ large enough.
This yields new examples of Riemann surfaces and domains of holomorphy in 
${{\mathbb{C}}^{n}}$ with corona. We consider the important special case when $Y$ is the unit ball 
$\mathbb{B}$ in ${{\mathbb{C}}^{n}}$ , and show that for $d$ large enough, every bounded holomorphic function on $X$ extends to a unique function in the intersection of all the nontrivial weighted Bergman spaces on $\mathbb{B}$ . Finally, assuming that the covering group is arithmetic, we establish three dichotomies concerning the extension of bounded holomorphic and harmonic functions from $X$ to $\mathbb{B}$ .
