  We study the dynamics of a generic endomorphism $f$ of an Oka-Stein manifold
$X$. Such manifolds include all connected linear algebraic groups and, more
generally, all Stein homogeneous spaces of complex Lie groups. We give several
descriptions of the Fatou set and the Julia set of $f$. In particular, we show
that the Julia set is the derived set of the set of attracting periodic points
of $f$ and that it is also the closure of the set of repelling periodic points
of $f$. Among other results, we prove that $f$ is chaotic on the Julia set and
that every periodic point of $f$ is hyperbolic. We also give an explicit
description of the "Conley decomposition" of $X$ induced by $f$ into
chain-recurrence classes and basins of attractors. For $X=\mathbb{C}$, we prove
that every Fatou component is a disc and that every point in the Fatou set is
attracted to an attracting cycle or lies in a dynamically bounded wandering
domain (whether such domains exist is an open question).
