Simultaneous linearization and centralizers of parabolic self-maps II: positive hyperbolic step

Abstract

The study of holomorphic self-mappings of the unit disc commuting under the composition goes back to A.L. Shields (1964), W.A. Pranger (1970), D.F. Behan (1973), and C.C. Cowen (1984). In many situations, the centralizer of a holomorphic self-map $\phi :\mathbb{D}\to \mathbb{D}$, i.e. the semigroup ${\mathcal{Z}}_{\forall }(\phi ):=\{\psi \in \mathsf{Hol}(\mathbb{D}):\psi \circ \phi =\phi \circ \psi \}$ turns out to be commutative. However, this does not hold for the case of a parabolic self-map $\phi$ of positive hyperbolic step, which is analyzed in detail in this paper. We investigate the relationships among commutativity, simultaneous linearization, and holomorphic models. In particular, we obtain existence and uniqueness results for the simultaneous linearization of commuting pairs $\phi$, $\psi \in {\mathcal{Z}}_{\forall }(\phi )$. Furthermore, extending this notion to arbitrary families of holomorphic self-mappings, we show that a given (finite or infinite) family in the centralizer $\Delta \subset {\mathcal{Z}}_{\forall }(\phi )$ can be simultaneously linearized together with$\phi$ if and only if any two elements of$\Delta$ commute with each other. This gives a far reaching extension of Cowen's result concerning commuting pairs in$\mathsf{Hol}(\mathbb{D})$.