Using the Stone-Čech compactification
β
Z
\beta \mathbf Z
of integers, we introduce a free extension of an almost periodic flow. Together with some properties of outer functions, we see that, in a certain class of ergodic Hardy spaces
H
p
(
μ
)
H^p(\mu )
,
1
≤
p
≤
∞
1\le p\le \infty
, the corresponding subspaces
H
0
p
(
μ
)
H_0^p(\mu )
are all singly generated. This shows the existence of maximal weak-
∗
^*
Dirichlet algebras, different from
H
∞
H^\infty
of the disc, for which the single generator problem is settled.