Dynamics of semiregular polynomial maps
type: Item
instance of: scholarly article
title: Dynamics of semiregular polynomial maps
zbMATH Open document ID: 1059.37033
DOI: 10.1007/BF02432910
author: Tien-Cuong Dinh
author: Nessim Sibony
published in: Arkiv för Matematik
publication date: 2004-09-28T00:00:00Z
review text: Let \(f = (P_1,\ldots,P_k)\) be a proper polynomial map of \(\mathbb{C}^k\) and \(d_1 \geq \ldots \geq d_k\) be the degrees of the coordinates. We note \(\alpha\) the escape function rate on \(\mathbb{C}^k\), defined as \(\alpha(z) := \limsup {1 \over n} \log^+ \log^+ abs { f^n (z) }\). Observe that if \(f\) extends holomorphically to the projective space \(\mathbb{P}^k\), then \(\alpha\) takes its value in \(\{0,d\}\), where \(d\) is the common degree of the \(P_i\). In that situation, one may introduce the Green function, the Green current and the equilibrium measure of \(f\) [see \textit{N. Sibony} in: D. Cerveau et al., Dynamique et géométrie complexes. Paris: Société Mathématique de France. Panor. Synth. 8, 97--185 (1999; Zbl 1020.37026)]. The article deals with the more general case of ``regular maps''. The function \(\alpha\) then takes its value in \(\{0,d_1,\ldots,d_k\}\). It is also possible to build the equilibrium measure \(\mu\), by considering the partial Green functions defined on the closed subset \(\{ \alpha \leq d_i \}\). The proof relies on analysis of psh functions with respect to closed positive currents. The authors give also a lower estimate of the dimension of \(\mu\) in terms of the topological degree and the topological Lyapounov exponent of \(f\). Note that the preceding result remain valid for the larger class of ``semi-regular maps'', which are regular maps composed at the source with a projective polynomial map.
reviewed by: Christophe Dupont
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 32H50
zbMATH DE Number: 2103839
zbMATH Keywords: holomorphic dynamics
zbMATH Keywords: equilibrium measure
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02432910
OpenAlex ID: W2045209186
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description: scientific article; zbMATH DE number 2103839