In this talk, we first review recent advances in characterizing the reducing subspaces of multiplication operators induced by several classes of inner functions on the Bergman space over the unit disk.we develop new techniques to analyze the analytic continuations of local inverses of a finite Blaschke product B by detecting the critical points of B. We introduce the notion of terminal branched components of B in a uniform way to classify reducing subspaces of the multiplication operators by Blaschke products with lower order on the Bergman space.
