Dynamical compactifications of \(\mathbb{C}^2\)
type: Item
instance of: scholarly article
title: Dynamical compactifications of \(\mathbb{C}^2\)
zbMATH Open document ID: 1244.32012
author: Charles Favre
author: Mattias Jonsson
published in: Annals of Mathematics. Second Series
publication date: 2011-10-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/0711.2770
review text: In this very interesting paper the authors use the full force of the dynamical theory of valuations they developed in the last few years (see, in particular, [the authors, ``Eigenvaluations'', Ann. Sci. Éc. Norm. Supér. (4) 40, No. 2, 309--349 (2007; Zbl 1135.37018)], and [\textit{S. Boucksom} and the authors, ``Degree growth of meromorphic surface maps'', Duke Math. J. 141, No. 3, 519--538 (2008; Zbl 1185.32009)]) to prove several deep theorems about the dynamics of polynomial endomorphisms of \(\mathbb{C}^2\). The main result is the following Theorem A. Let \(F:\mathbb{C}^2\to\mathbb{C}^2\) be a polynomial mapping. Then there exists a projective compactification \(X\supset\mathbb{C}^2\) with at most quotient singularities such that the lift \(\tilde F\) of \(F\) to \(X\) is eventually algebraically stable, that is, there is \(n\geq 1\) such that \((\tilde{F}^{n+j} )^* = (\tilde{F}^n )^* (\tilde{F}^j )^* = (\tilde{F}^n )^* (\tilde{F}^*)^j\) on the Picard group Pic\((X)\) for all \(j\geq 1\). Furthermore, in many cases (actually always up to replacing \(F\) by \(F^2\)) the compactification \(X\) is smooth, and it is obtained from \(\mathbb{P}^2\) with finitely many blow-ups at infinity. A first non-trivial consequence of this theorem is the fact that the sequence \(\{\deg(F^n)\}_{n\geq0}\) of degrees of the iterates of \(F\) satisfies an integral linear recursion formula; this should be compared with the examples due to \textit{B. Hasselblatt} and \textit{J. Propp} [``Degree-growth of monomial maps'', Ergodic Theory Dyn. Syst. 27, No. 5, 1375--1397 (2007; Zbl 1143.37032)] of rational self-maps of \(\mathbb{C}^2\) such that the sequence of degrees does not satisfy any linear recursion formula. The limit of \(\deg(F^n)^{1/n}\) as \(n\to\infty\) is the asymptotic degree \(\lambda_1\) of \(F\), and it is known that \(\lambda^2_1\geq\lambda_2\) always, where \(\lambda_2\) is the topological degree of \(F\). When \(\lambda^2_1 > \lambda_2\), it was already proved in [Bouc…
reviewed by: Marco Abate
Mathematics Subject Classification ID: 32H50
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 32H04
Mathematics Subject Classification ID: 32J05
Mathematics Subject Classification ID: 37F10
zbMATH DE Number: 5960659
zbMATH Keywords: algebraic stability
zbMATH Keywords: compactifications
zbMATH Keywords: degree growth
zbMATH Keywords: holomorphic dynamics
zbMATH Keywords: polynomial mappings
zbMATH Keywords: Riemann-Zariski spaces
MaRDI profile type: Publication
OpenAlex ID: W2962798991
arXiv ID: 0711.2770
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DOI: 10.4007/ANNALS.2011.173.1.6
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