Critical structures of inner functions II

Abstract

We study the correspondence proposed in \cite{critical-structures} between inner functions modulo post-compositions with automorphisms of the unit disk and cyclic subspaces of the weighted Bergman space $A_1^2$. The correspondence sends an inner function $I$ to the invariant subspace $[I^{\prime }]$, and in the opposite direction, assigns to a non-zero function $H\in A_1^2$ the Liouville map $I_H$ associated to the canonical solution of the Gauss curvature equation $\Delta u=|H{|}^2{e}^{2u}$. We prove that $I_H^{\prime }$ generates the same cyclic subspace as $H$. Combined with the results in \cite{critical-structures}, this shows that these two mappings are inverses of one another, and hence the correspondence $I\to [I^{\prime }]$ is a bijection.

11 pages