  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 $C_\phi$
induced by non-elliptic automorphisms. This description exhibits a surprising
connection to model spaces. Finally, we give a constructive characterization of
the $C_\phi$-eigenfunctions in $H^p$ for $1\le p\le \infty$.
