Orbits of non-elliptic disc automorphisms on

Motivated by the Invariant Subspace Problem, we describe explicitly the closed subspace H 2 generated by the limit points in the H 2 norm of the orbit of a thin Blaschke product B under composition operators C φ induced by non-elliptic automorphisms. This description exhibits a surprising connection to model spaces. Finally, we give a constructive characterization of the C φ -eigenfunctions in H p for 1 ⩽ p ⩽ ∞ .