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 \( K_f:= \overline{\textrm{span} \{f, C_{\phi _a}f, C^2_{\phi _a}f, \ldots \}}\) 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 \(f'\) 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.
