JournalsrmiVol. 26, No. 1pp. 57–76

Valiron’s construction in higher dimension

  • Filippo Bracci

    Università di Roma 'Tor Vergata', Italy
  • Graziano Gentili

    Università degli Studi di Firenze, Italy
  • Pietro Poggi-Corradini

    Kansas State University, Manhattan, USA
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We consider holomorphic self-maps φ\varphi of the unit ball BN\mathbb B^N in CN\mathbb C^N (N=1,2,3,N=1,2,3,\dots). In the one-dimensional case, when φ\varphi has no fixed points in D\defeqB1\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 φ\varphi, and therefore, in this case, the dynamical properties of φ\varphi are well understood. In what follows, we generalize the classical Valiron construction to higher dimensions under some weak assumptions on φ\varphi at its Denjoy-Wolff point. As a result, we construct a semi-conjugation σ\sigma, which maps the ball into the right half-plane of C\mathbb C, and solves the functional equation σφ=λσ\sigma\circ \varphi=\lambda \sigma, where λ>1\lambda > 1 is the (inverse of the) boundary dilation coefficient at the Denjoy-Wolff point of φ\varphi.

Cite this article

Filippo Bracci, Graziano Gentili, Pietro Poggi-Corradini, Valiron’s construction in higher dimension. Rev. Mat. Iberoam. 26 (2010), no. 1, pp. 57–76

DOI 10.4171/RMI/593