Let
G
G
be a countably infinite group, and let
μ
\mu
be a generating probability measure on
G
G
. We study the space of
μ
\mu
-stationary Borel probability measures on a topological
G
G
space, and in particular on
Z
G
Z^G
, where
Z
Z
is any perfect Polish space. We also study the space of
μ
\mu
-stationary, measurable
G
G
-actions on a standard, nonatomic probability space.
Equip the space of stationary measures with the weak* topology. When
μ
\mu
has finite entropy, we show that a generic measure is an essentially free extension of the Poisson boundary of
(
G
,
μ
)
(G,\mu )
. When
Z
Z
is compact, this implies that the simplex of
μ
\mu
-stationary measures on
Z
G
Z^G
is a Poulsen simplex. We show that this is also the case for the simplex of stationary measures on
{
0
,
1
}
G
\{0,1\}^G
.
We furthermore show that if the action of
G
G
on its Poisson boundary is essentially free, then a generic measure is isomorphic to the Poisson boundary.
Next, we consider the space of stationary actions, equipped with a standard topology known as the weak topology. Here we show that when
G
G
has property (T), the ergodic actions are meager. We also construct a group
G
G
without property (T) such that the ergodic actions are not dense, for some
μ
\mu
.
Finally, for a weaker topology on the set of actions, which we call the very weak topology, we show that a dynamical property (e.g., ergodicity) is topologically generic if and only if it is generic in the space of measures. There we also show a Glasner-King type 0-1 law stating that every dynamical property is either meager or residual.