Shift invariant subspaces in growth spaces and sets of finite entropy

We investigate certain classes of shift invariant subspaces in growth spaces on the unit disc of the complex plane determined by a majorant $w$, which include the classical Korenblum growth spaces. Our main result provides a complete description of shift invariant subspaces generated by Nevanlinna class functions in growth spaces, where we show that they are of Beurling-type. In particular, our result generalizes the celebrated Korenblum-Roberts Theorem. It turns out that singular inner functions play the decisive role in our description, phrased in terms of certain $w$-entropy conditions on the carrier sets of the associated singular measures, which arise in connection to boundary zero sets for analytic functions in the unit disc having modulus of continuity not exceeding $w$ on the unit circle. Furthermore, this enables us to establish an intimate link between shift invariant subspace generated by inner functions and the containment of the above mentioned analytic function spaces in the corresponding model spaces.