Let \(\varphi :{\mathbb {D}} \rightarrow {\mathbb {D}}\) be a parabolic self-map of the unit disc \({\mathbb {D}}\) having zero hyperbolic step. We study holomorphic self-maps of \({\mathbb {D}}\) commuting with \(\varphi \) . In particular, we answer a question from Gentili and Vlacci (1994) by proving that \(\psi \in \mathsf {Hol({\mathbb {D}},{\mathbb {D}})}\) commutes with \(\varphi \) if and only if the two self-maps have the same Denjoy – Wolff point and \(\psi \) is a pseudo-iterate of \(\varphi \) in the sense of Cowen. Moreover, we show that the centralizer of \(\varphi \) , i.e. the semigroup \({\mathscr {Z}}_\forall (\varphi ):=\{\psi \in \mathsf {Hol({\mathbb {D}},{\mathbb {D}})}:\psi \circ \varphi =\varphi \circ \psi \}\) is commutative. We also prove that if \(\varphi \) is univalent, then all elements of \({\mathscr {Z}}_\forall (\varphi )\) are univalent as well, and if \(\varphi \) is not univalent, then the identity map is an isolated point of \({\mathscr {Z}}_\forall (\varphi )\) . The main tool is the machinery of simultaneous linearization, which we develop using holomorphic models for iteration of non-elliptic self-maps originating in works of Cowen and Pommerenke.
