Let
H
0
(
M
n
,
μ
)
{\mathcal {H}_0}({M^n},\mu )
denote the set of all homeomorphisms of a compact manifold
M
n
{M^n}
that preserve a locally positive nonatomic Borel probability measure
μ
\mu
and are isotopic to the identity. The notion of the mean rotation vector for a torus homeomorphism has been extended by Fathi to a continuous map
θ
\theta
on
H
0
(
M
n
,
μ
)
{\mathcal {H}_0}({M^n},\mu )
. We show that any abstract ergodic behavior typical for automorphisms of
(
M
n
,
μ
)
({M^n},\mu )
as a Lebesgue space is also typical not only in
H
0
(
M
n
,
μ
{\mathcal {H}_0}({M^n},\mu
but also in each closed subset of constant
θ
\theta
. By typical we mean dense
G
δ
{G_\delta }
in the appropriate space. Weak mixing is an example of such a typical abstract ergodic behavior. This contrasts sharply with a deep result of the KAM theory that for some rotation vectors
v
→
\overrightarrow v
, there is an open neighborhood of rotation by
v
→
\overrightarrow v
, in the space of smooth volume preserving
n
n
-torus diffeomorphisms with
θ
=
v
→
\theta = \overrightarrow v
, where each diffeomorphism in the open set is conjugate to rotation by
v
→
\overrightarrow v
(and hence cannot be weak mixing).