Simultaneous linearization and centralizers of parabolic self-maps I: zero hyperbolic step

2025/08/04 by Contreras, Manuel D., Díaz-Madrigal, Santiago, Gumenyuk, Pavel
#30C55 #30D05 #37F44 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.02809

Abstract

Let φ:\mathbb D → \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\inHol(\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_∀(φ):=\ψ:ψ∘φ=φ∘ψ\ is commutative. We also prove that if φ is univalent, then all elements of \mathcal Z_∀(φ) are univalent as well, and if φ is not univalent, then the identity map is an isolated point of \mathcal Z_∀(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.

Citations

• Simultaneous linearization and centralizers of parabolic self-maps II: positive hyperbolic step
• Hyperbolic distortion and conformality at the boundary
• Criteria for extension of commutativity to fractional iterates of holomorphic self-maps in the unit disc
• Centralizers of non-elliptic univalent self-maps and the embeddability problem in the unit disc
• Semigroup-fication of univalent self-maps of the unit disc
• On the K "onigs function of semigroups of holomorphic self-maps of the\n unit disc
• Simultaneous models for commuting holomorphic self-maps of the ball
• Canonical Models for holomorphic iteration
• Simultaneous linearization of holomorphic germs in presence of resonances
• On the uniqueness of classical semiconjugations for self-maps of the disk

Related

• Simultaneous linearization and centralizers of parabolic self-maps II: positive hyperbolic step
• Centralizers of non-elliptic univalent self-maps and the embeddability problem in the unit disc
• Semigroup-fication of univalent self-maps of the unit disc
• Criteria for extension of commutativity to fractional iterates of holomorphic self-maps in the unit disc
• Simultaneous models for commuting holomorphic self-maps of the ball