In this paper we generalize Beurling’s invariant subspace theorem to the Hardy classes on a Riemann surface with infinite handles. The problem is to classify all closed (
weak
∗
\text {weak}^{\ast }
closed, if
p
=
∞
p = \infty
)
H
∞
(
d
χ
)
{H^\infty }(d\chi )
-submodules, say
m
\mathfrak {m}
, of
L
p
(
d
χ
)
{L^p}(d\chi )
,
1
⩽
p
⩽
∞
1 \leqslant p \leqslant \infty
, where
d
χ
d\chi
is the harmonic measure on the Martin boundary of a Riemann surface
R
R
, and
H
∞
(
d
χ
)
{H^\infty }(d\chi )
is the set of boundary functions of all bounded analytic functions on
R
R
. Our main result is stated roughly as follows. Let
R
R
be of Parreau-Widom type, that is, the space
H
∞
(
R
,
γ
)
{H^\infty }(R,\gamma )
of bounded analytic sections contains a nonzero element for every complex flat line bundle
γ
∈
π
(
R
)
∗
\gamma \in \pi {(R)^{\ast }}
. We may assume, without loss of generality, that the Green’s function of
R
R
vanishes at the infinity. Set
m
∞
(
γ
)
=
sup
{
|
f
(
O
)
|
:
f
∈
H
∞
(
R
,
γ
)
,
|
f
|
⩽
1
}
{m^\infty }(\gamma ) = \sup \{ |f({\mathbf {O}})|:f \in {H^\infty }(R,\gamma ),|f| \leqslant 1\}
for a fixed point
O
{\mathbf {O}}
of
R
R
. Then, a necessary and sufficient condition in order that every such an
m
\mathfrak {m}
takes either the form
m
=
C
E
L
p
(
d
χ
)
\mathfrak {m} = {C_E}{L^p}(d\chi )
, where
C
E
{C_E}
is the characteristic function of a set
E
E
, or the form
m
=
q
H
p
(
d
χ
,
γ
)
\mathfrak {m} = q{H^p}(d\chi ,\gamma )
, where
|
q
|
=
1
|q| = 1
a.e. and
γ
\gamma
is some element of
π
(
R
)
∗
\pi {(R)^{\ast }}
is that
m
∞
(
γ
)
{m^\infty }(\gamma )
is continuous for the variable
γ
∈
π
(
R
)
∗
\gamma \in \pi {(R)^{\ast }}
.