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 $M_phi$ on the Dirichlet space with symbol of finite Blaschke product. The reducing subspaces of $M_phi$ 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 $M_phi$ on the Bergman space, and we discover a new way to study the Riemann surface for $phi^{-1}circphi$. By this means, we determine the reducing subspaces of $M_phi$ on the Dirichlet space when the order of $phi$ is $5$; $6$; $7$ and answer some questions of Douglas-Putinar-Wang cite{DPW12}.