  For any countable group $\Gamma$ satisfying the ``weak Rohlin property'', and
for any dynamical property, the set of $\Gamma$-actions with that property is
either residual or meager. The class of groups with the weak Rohlin property
includes each lattice $\integers^{\times{d}}$; indeed, all countable discrete
amenable groups.
  For $\Gamma$ an arbitrary countable group, let $\actsp$ be the set of
$\Gamma$-actions on the unit circle $Y$. We establish an Equivalence theorem by
showing that a dynamical property is Baire/meager/residual in $\actsp$ if and
only if it is Baire/meager/residual in the set of shift-invariant measures on
the product space $Y^{\times\Gamma}$.
