Open Access
March, 2010 Valiron's construction in higher dimension
Filippo Bracci , Graziano Gentili , Pietro Poggi-Corradini
Rev. Mat. Iberoamericana 26(1): 57-76 (March, 2010).


We consider holomorphic self-maps $\varphi$ of the unit ball $\mathbb B^N$ in $\mathbb C^N$ ($N=1,2,3,\dots$). In the one-dimensional case, when $\varphi$ has no fixed points in $\mathbb D\defeq \mathbb B^1$ and is of hyperbolic type, there is a classical renormalization procedure due to Valiron which allows to semi-linearize the map $\varphi$, and therefore, in this case, the dynamical properties of $\varphi$ are well understood. In what follows, we generalize the classical Valiron construction to higher dimensions under some weak assumptions on $\varphi$ at its Denjoy-Wolff point. As a result, we construct a semi-conjugation $\sigma$, which maps the ball into the right half-plane of $\mathbb C$, and solves the functional equation $\sigma\circ \varphi=\lambda \sigma$, where $\lambda > 1$ is the (inverse of the) boundary dilation coefficient at the Denjoy-Wolff point of $\varphi$.


Download Citation

Filippo Bracci . Graziano Gentili . Pietro Poggi-Corradini . "Valiron's construction in higher dimension." Rev. Mat. Iberoamericana 26 (1) 57 - 76, March, 2010.


Published: March, 2010
First available in Project Euclid: 16 February 2010

zbMATH: 1198.32006
MathSciNet: MR2666307

Primary: 32A10‎ , 32H50
Secondary: 30D05

Keywords: dynamics of holomorphic self-maps , hyperbolic maps , intertwining maps , iteration theory , Linearization

Rights: Copyright © 2010 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.26 • No. 1 • March, 2010
Back to Top