Beurling and Model subspaces invariant under a universal operator

In this article, we characterize the Beurling and Model subspaces of the Hardy-Hilbert space $H^2(\mathbb{D})$ invariant under the composition operator $C_{{\phi }_a}f=f\circ {\phi }_a$, where ${\phi }_a(z)=az+1-a$ for $a\in (0,1)$ is an affine self-map of the open unit disk $\mathbb{D}$. These operators have universal translates (in the sense of Rota) and have attracted attention recently due to their connection with the Invariant Subspace Problem (ISP) and the classical Cesàro operator.