For
f
∈
ℓ
∞
(
N
)
f\in \ell ^{\infty }( \mathbb {N})
let
T
f
Tf
be defined by
T
f
(
n
)
=
1
n
∑
i
=
1
n
f
(
i
)
Tf(n)=\frac {1}{n}\sum _{i=1}^{n}f(i)
. We investigate permutations
g
g
of
N
\mathbb {N}
, which satisfy
T
f
(
n
)
−
T
f
g
(
n
)
→
0
Tf(n)-Tf_{g}(n)\to 0
as
n
→
∞
n\to \infty
with
f
g
(
n
)
=
f
(
g
n
)
f_{g}(n)=f(gn)
for
f
∈
ℓ
∞
(
N
)
f\in \ell ^{\infty }( \mathbb {N})
(i.e.
g
g
is in the Lévy group
G
)
\mathcal {G})
, or for
f
f
in the subspace of Cesàro-summable sequences. Our main interest are
G
\mathcal {G}
-invariant means on
ℓ
∞
(
N
)
\ell ^{\infty }( \mathbb {N})
or equivalently
G
\mathcal {G}
-invariant probability measures on
β
N
\beta \mathbb {N}
. We show that the adjoint
T
∗
T^{*}
of
T
T
maps measures supported in
β
N
∖
N
\beta \mathbb {N} \setminus \mathbb {N}
onto a weak*-dense subset of the space of
G
\mathcal {G}
-invariant measures. We investigate the dynamical system
(
G
,
β
N
)
( \mathcal {G}, \beta \mathbb {N})
and show that the support set of invariant measures on
β
N
\beta \mathbb {N}
is the closure of the set of almost periodic points and the set of non-topologically transitive points in
β
N
∖
N
\beta \mathbb {N}\setminus \mathbb {N}
. Finally we consider measures which are invariant under
T
∗
T^{*}
.