×

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)

MSC:

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
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] Abate, M.: Iteration theory of holomorphic maps on taut manifolds Mediterranean Press, Rende, Cosenza, 1989. · Zbl 0747.32002
[2] Bayart, F.: The linear fractional model on the ball. Rev. Mat. Iberoam. 24 (2008), no. 3, 765-824. · Zbl 1165.32009
[3] Bracci, F. and Gentili, G.: Solving the Schröder equation at the boundary in several variables. Michigan Math. J. 53 (2005), no. 5, 337-356. · Zbl 1130.32007
[4] Bracci, F. and Poggi-Corradini, P.: On Valiron’s theorem. In Future Trends in Geometric Function Theory , 39-55. Rep. Univ. Jyväskylä Dep. Math. Stat. 92 . Univ. Jyväskylä, Jyväskylä, 2003. · Zbl 1043.30010
[5] Bourdon, P. and Shapiro, J.: Cyclic phenomena for composition operators. Mem. Amer. Math. Soc. 125 (1997), no. 596. · Zbl 0996.47032
[6] Cowen, C.C.: Iteration and the solution of functional equations for functions analytic in the unit disk. Trans. Amer. Math. Soc. 265 (1981), no. 1, 69-95. JSTOR: · Zbl 0476.30017
[7] D’Angelo, J.P.: Several complex variables and the geometry of real hypersurfaces. Studies in Advanced Math. CRC Press, Boca Raton, FL, 1993. · Zbl 0854.32001
[8] De Fabritiis, C. and Gentili, G.: On holomorphic maps which commute with hyperbolic automorphisms. Adv. Math. 144 (1999), no. 2, 119-136. · Zbl 0976.32014
[9] Kobayashi, S.: Hyperbolic complex spaces. Grundlehren der Mathematischen Wissenschaften 318 . Springer-Berlag, Berlin, 1998. · Zbl 0917.32019
[10] MacCluer, B.D.: Iterates of holomorphic self-maps of the unit ball in \(\mathbbC^N\). Michigan Math. J. 30 (1983), no. 1, 97-106. · Zbl 0528.32019
[11] Pommerenke, Ch.: On the iteration of analytic functions in a halfplane, I. J. London Math. Soc. (2) 19 (1979), no. 3, 439-447. · Zbl 0398.30014
[12] Pommerenke, Ch.: On asymptotic iteration of analytic functions in the disk. Analysis 1 (1981), no. 1, 45-61. · Zbl 0477.30019
[13] Rudin, W.: Function theory in the unit ball of \(\mathbbC^n\). Grundlehren der Mathematischen Wissenschaften 241 . Springer-Berlag, New York-Berlin, 1980. · Zbl 0495.32001
[14] Valiron, G.: Sur l’iteration des fonctions holomorphes dans un demi-plan. Bull. Sci. Math. (2) 55 (1931), 105-128. · JFM 57.0381.03
[15] Valiron, G.: Fonctions Analytiques. Presses Universitaires de France, Paris, 1954. · Zbl 0055.06702
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.