Orbits of non-elliptic disc automorphisms on Hp

Motivated by the Invariant Subspace Problem, we describe explicitly the closed subspace H2 generated by the limit points in the H2 norm of the orbit of a thin Blaschke product B under composition operators Cφ induced by non-elliptic automorphisms. This description exhibits a surprising connection to model spaces. Finally, we give a constructive characterization of the Cφ-eigenfunctions in Hp for 1⩽p⩽∞.