Beurling type invariant subspaces of composition operators

Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$, let $H^2$ denote the Hardy space on $\mathbb{D}$ and let $\varphi : $\mathbb{D}$ \rightarrow $\mathbb{D}$$beaholomorphicselfmapof$$\mathbb{D}$$\mathrm{.}Thecompositionoperator$C_{\varphi}$on$H^2$ is defined by $$(C_{\phi }f)(z)=f(\phi (z))(f\in H^2,z\in \mathbb{D})\mathrm{.}$$ Denote by $\mathcal{S}(\mathbb{D})$ the set of all functions that are holomorphic and bounded by one in modulus on $\mathbb{D}$, that is $$\mathcal{S}(\mathbb{D})=\{\psi \in H^{\mathrm{\infty }}(\mathbb{D}):\mathrm{\parallel }\psi {\mathrm{\parallel }}_{\mathrm{\infty }}:=\underset{z\in \mathbb{D}}{\sup }\mathrm{\mid }\psi (z)\mathrm{\mid }\le 1\}\mathrm{.}$$ The elements of $\mathcal{S}(\mathbb{D})$ are called Schur functions. The aim of this paper is to answer the following question concerning invariant subspaces of composition operators: Characterize $\phi$, holomorphic self maps of $\mathbb{D}$, and inner functions $\theta \in H^\infty($\mathbb{D}$)$suchthattheBeurlingtypeinvariantsubspace$\theta H^2$isaninvariantsubspacefor$C_{\varphi}$\mathrm{.}Weprovethefollowingresult:$C_{\varphi} (\theta H^2) \subseteq \theta H^2$ if and only if $$\frac{\theta \circ \phi }{\theta }\in \mathcal{S}(\mathbb{D})\mathrm{.}$$ This classification also allows us to recover or improve some known results on Beurling type invariant subspaces of composition operators.