Invariant subspaces on Riemann surfaces of Parreau-Widom type

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$^{\ast}$ closed, if $p = \infty$) ${H^\infty }(d\chi )$-submodules, say $\mathfrak{m}$, of ${L^p}(d\chi )$, $1 \leqslant p \leqslant \infty$, where $d\chi$ is the harmonic measure on the Martin boundary of a Riemann surface $R$, and ${H^\infty }(d\chi )$ is the set of boundary functions of all bounded analytic functions on $R$. Our main result is stated roughly as follows. Let $R$ be of Parreau-Widom type, that is, the space ${H^\infty }(R,\gamma )$ of bounded analytic sections contains a nonzero element for every complex flat line bundle $\gamma \in \pi {(R)^{\ast}}$. We may assume, without loss of generality, that the Green's function of $R$ vanishes at the infinity. Set ${m^\infty }(\gamma ) = \sup \{ \vert f({\mathbf{O}})\vert:f \in {H^\infty }(R,\gamma ),\vert f\vert \leqslant 1\}$ for a fixed point ${\mathbf{O}}$ of $R$. Then, a necessary and sufficient condition in order that every such an $\mathfrak{m}$ takes either the form $\mathfrak{m} = {C_E}{L^p}(d\chi )$, where ${C_E}$ is the characteristic function of a set $E$, or the form $\mathfrak{m} = q{H^p}(d\chi ,\gamma )$, where $\vert q\vert = 1$ a.e. and $\gamma$ is some element of $\pi {(R)^{\ast}}$ is that ${m^\infty }(\gamma )$ is continuous for the variable $\gamma \in \pi {(R)^{\ast}}$.