Valiron’s construction in higher dimension. (English) Zbl 1198.32006

In 1931, G. Valiron [Bull. Sci. Math., II. Ser. 55, 105–128 (1931; Zbl 0001.28101; JFM 57.0381.03)] proved that if \(\varphi:\Delta\to\Delta\) is a holomorphic self-map of the unit disk \(\Delta\subset{\mathbb C}\) with Wolff-Denjoy point \(\tau\in\partial\Delta\) and boundary dilatation coefficient \(c\) in \(\tau\) strictly between 0 and 1 (the so-called hyperbolic case, excluding only the case \(c=1\)) then there exists a non-constant holomorphic map \(\theta:\Delta\to H\) (where \(H\subset{\mathbb C}\) is the right half-plane) semiconjugating \(\varphi\) with the multiplication by \(c^{-1}\), i.e., such that \(\theta\circ\varphi=c^{-1}\theta\). Valiron built \(\theta\) working, via the Cayley transform \(C\), in \(H\) and proved that \(\theta\) is the limit of the sequence \(x_k^{-1}\phi^k\), where \(\phi=C\circ\varphi\circ C^{-1}\) and \(x_k=\text{Re}\,\phi^k(1)\).
In this paper, the authors show that a version of Valiron’s construction works in the unit ball \(B^n\subset{\mathbb C}^n\) under suitable assumptions. The main statement is the following: let \(\varphi: B^n\to B^n\) be a holomorphic self-map with Wolff-Denjoy point \(\tau\in\partial B^n\) and boundary dilatation coefficient \(c=\liminf_{z\to\tau}(1-\|\varphi(z)\|)/(1-\|z\|)\in(0,1)\); again, the only excluded case is \(c=1\). Assume that there exists \(z_0\in B^n\) such that, setting \(z_k=\varphi^k(z_0)\), one has \(k_{B^n}(z_k,\langle z_k,\tau\rangle\tau)\to 0\) as \(k\to+\infty\) (where \(k_{B^n}\) denotes the Kobayashi distance of \(B^n\)), and that the ratio \((1-\langle\varphi(z), \tau\rangle)/(1-\langle z,\tau\rangle)\) admits a finite \(K\)-limit at \(\tau\) (where \(\langle\cdot\,,\cdot \rangle\) denotes the canonical Hermitian product). Then one can apply an \(n\)-dimensional version of Valiron’s construction to find \(\theta: B^n\to H\) such that \(\theta\circ\varphi=c^{-1}\theta\).
Reviewer: Marco Abate (Pisa)


32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
32A10 Holomorphic functions of several complex variables
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
32A30 Other generalizations of function theory of one complex variable
Full Text: DOI arXiv Euclid


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