Green currents for quasi-algebraically stable meromorphic self-maps of \(\mathbb{P}^k\)
type: Item
instance of: scholarly article
title: Green currents for quasi-algebraically stable meromorphic self-maps of \(\mathbb{P}^k\)
zbMATH Open document ID: 1297.37023
DOI: 10.5565/PUBLMAT_56112_05
author: Viêt Anh Nguyên
published in: Publicacions Matemàtiques
publication date: 2012-07-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/0902.1376
reviewed by: Filippo Bracci
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 32U40
Mathematics Subject Classification ID: 32H50
Mathematics Subject Classification ID: 37F50
zbMATH DE Number: 6055945
zbMATH Keywords: quasi-algebraically stable meromorphic maps
zbMATH Keywords: dynamics on the projective space
zbMATH Keywords: Green current
zbMATH Keywords: algebraic degree
MaRDI profile type: Publication
arXiv ID: 0902.1376
OpenAlex ID: W1979070205
review text: Let \(f: \mathbb P^k \to \mathbb P^k\) be a dominant meromorphic map. The map \(f\) is algebraically stable if the algebraic degree of \(f^{\circ n}\) equals the algebraic degree of \(f\), \(d(f)\), to the power \(n\). For algebraically stable maps many things are known. For instance Sibony proved that if \(f\) is algebraically stable and \(d(f)>1\), then there exists a unique current \(T\), called the Green current of \(f\), which satisfies \(f^\ast(T)=d(f)T\) and essentially describes the dynamics of \(f\). For non-algebraically stable mappings, not much is known. Some classes of non algebraically stable maps have been studied by Bonifant and Fornæss, while Diller and Favre, and Favre, Jonsson and later together with Boucksom, have studied non algebraically stable maps in dimension two (references are in the bibliography). Other particular cases are studied by Hasselblatt and Propp and by Bedford and Kim.NEWLINENEWLINEIn the paper under review the author studies a new class of non algebraically stable maps: the class of ``quasi-algebraically stable maps''. A map is quasi-algebraically stable provided there exist an integer \(n_0\geq 1\), a homogeneous polynomial \(H\) and a sequence \(\{F_n\}\) of liftings to \(\mathbb C^{k+1}\) of the sequence of iterates \(\{f^{\circ n}\}\) of \(f\) such that \(F_n=F_1\circ F_{n-1}\) for \(n\leq n_0\) and \(F_n=F_1\circ F_{n-1}/H\circ F_{n-n_0-1}\) for \(n>n_0\).NEWLINENEWLINELet \(f\) be quasi-algebraically stable. Let \(\lambda\) denote the first dynamical degree of \(f\) and let \(h\) be the degree of \(H\). It is known that \(\lambda\) is a root of the polynomial \(P(t)= t^{n_0+1}-dt^{n_0}+h\). Assume that \(\lambda\) is a simple root of \(P(t)\) and \(\lambda>1\). Then the main result of the paper is the following: the upper semicontinuous regularization of \(\limsup_{n\to\infty}\log\|F_n\|/d(f^n)\) exists and defines a plurisubharmonic function \(u\) on \(\mathbb C^{k+1}\). The positive closed \((1,1)\)-current \(T\) on \(…
description: scientific article; zbMATH DE number 6055945