Simultaneous linearization and centralizers of parabolic self-maps I: zero hyperbolic step
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arXiv
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| Natura: | Preprint |
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2025
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| author | Contreras, Manuel D. Díaz-Madrigal, Santiago Gumenyuk, Pavel |
| author_facet | Contreras, Manuel D. Díaz-Madrigal, Santiago Gumenyuk, Pavel |
| contents | Let $φ:\mathbb D \to \mathbb D$ 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 $φ$. In particular, we answer a question from Gentili and Vlacci (1994) by proving that $ψ\in\mathsf{Hol(\mathbb D,\mathbb D)}$ commutes with $φ$ if and only if the two self-maps have the same Denjoy-Wolff point and $ψ$ is a pseudo-iterate of $φ$ in the sense of Cowen. Moreover, we show that the centralizer of $φ$, i.e. the semigroup $\mathcal Z_\forall(φ):=\{ψ:ψ\circφ=φ\circψ\}$ is commutative. We also prove that if $φ$ is univalent, then all elements of $\mathcal Z_\forall(φ)$ are univalent as well, and if $φ$ is not univalent, then the identity map is an isolated point of $\mathcal Z_\forall(φ)$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02809 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simultaneous linearization and centralizers of parabolic self-maps I: zero hyperbolic step Contreras, Manuel D. Díaz-Madrigal, Santiago Gumenyuk, Pavel Complex Variables Dynamical Systems 30C55, 37F44, 30D05 Let $φ:\mathbb D \to \mathbb D$ 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 $φ$. In particular, we answer a question from Gentili and Vlacci (1994) by proving that $ψ\in\mathsf{Hol(\mathbb D,\mathbb D)}$ commutes with $φ$ if and only if the two self-maps have the same Denjoy-Wolff point and $ψ$ is a pseudo-iterate of $φ$ in the sense of Cowen. Moreover, we show that the centralizer of $φ$, i.e. the semigroup $\mathcal Z_\forall(φ):=\{ψ:ψ\circφ=φ\circψ\}$ is commutative. We also prove that if $φ$ is univalent, then all elements of $\mathcal Z_\forall(φ)$ are univalent as well, and if $φ$ is not univalent, then the identity map is an isolated point of $\mathcal Z_\forall(φ)$. 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. |
| title | Simultaneous linearization and centralizers of parabolic self-maps I: zero hyperbolic step |
| topic | Complex Variables Dynamical Systems 30C55, 37F44, 30D05 |
| url | https://arxiv.org/abs/2508.02809 |