Shadowing phenomenon for composition operators on the Hardy space $H^2(\mathbb{D})$

Let $\phi$ be a holomorphic self-map of the open unit disk $\mathbb{D}.$ In this article, we study the shadowing phenomenon for composition operators $C_{\phi}f=f\circ \phi$ on the Hardy space $H^2(\mathbb{D}).$ We mainly characterize all the composition operators induced by linear fractional self-maps of $\mathbb{D}$ that have the positive shadowing property.