Reducing subspaces for multiplication operators on the Dirichlet space through local inverses and Riemann surface

This paper is devoted to the study of reducing subspaces for multiplication operator $M_{\varphi }$ on the Dirichlet space with symbol of finite Blaschke product. The reducing subspaces of $M_{\varphi }$ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surface to study the reducing subspaces of $M_{\varphi }$ on the Bergman space, and we discover a new way to study the Riemann surface for ${\varphi }^{-1}\circ \varphi$. By this means, we determine the reducing subspaces of $M_{\varphi }$ on the Dirichlet space when the order of $\varphi$ is $5$; $6$; $7$ and answer some questions of Douglas-Putinar-Wang \cite{DPW12}.