Minimal invariant subspaces for an affine composition operator

The composition operator $C_{{\phi }_a}f=f\circ {\phi }_a$ on the Hardy-Hilbert space $H^2(\mathbb{D})$ with affine symbol ${\phi }_a(z)=az+1-a$ and $0<a<1$ has the property that the Invariant Subspace Problem for complex separable Hilbert spaces holds if and only if every minimal invariant subspace for $C_{{\phi }_a}$ is one-dimensional. These minimal invariant subspaces are always singly-generated $K_f:=\stackrel{‾}{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{f,C_{{\phi }_a}f,C_{{\phi }_a}^{2}f,\dots \}}$ for some $f\in H^2(\mathbb{D})$. In this article we characterize the minimal $K_f$ when $f$ has a nonzero limit at the point $1$ or if its derivative $f^{\mathrm{\prime }}$ is bounded near $1$. We also consider the role of the zero set of $f$ in determining $K_f$. Finally we prove a result linking universality in the sense of Rota with cyclicity.