Dynamical compactifications of \(\mathbb{C}^2\)
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 [Boucksom and the authors, loc. cit.] that \(\deg(F^n)\sim\lambda^n_1\); in this paper they study the case \(\lambda^2_1 = \lambda_2\), showing that then either \(\deg(F^n)\sim n\lambda^n_1\) or \(\deg(F^n)\sim\lambda^n_1\), and giving normal forms (up to conjugation by a polynomial automorphism of \(\mathbb{C}^2\)) for both cases. The authors also study ergodic theory of polynomial endomorphisms of \(\mathbb{C}^2\). When \(\lambda_2 > \lambda_1\), it is well known that an ergodic measure of maximal entropy can be defined as a limit of preimages of a generic point (see, e.g., [\textit{J. E. Fornæss} and \textit{N. Sibony}, Complex dynamics in higher dimension. II. Ann. Math. Stud. 137, 135--182 (1995; Zbl 0847.58059)]). When \(\lambda_2 < \lambda_1\) this construction fails; \textit{V. Guedj} [``Dynamics of polynomial mappings of \(\mathbb{C}^2\)'', Am. J. Math. 124, No. 1, 75--106 (2002; Zbl 1198.32007)]) suggested instead to first define two positive closed \((1, 1)\)-currents, invariant under pullback and under pushforward respectively, and then to build the ergodic measure of maximal entropy by taking their intersection. The existence of the two currents in this context is another consequence of Theorem A; however, to ensure the existence of the intersection one needs a good control of the singularities of their potentials. As a final result, the authors are able to obtain such a control for the pullback invariant current, defined as \( \text{dd}^c G^+\), where \(G^+ = \lim_{n\to\infty}\lambda_1^{-n}\log^+\|F^n\|\). (It is not yet clear what should be the right construction of an ergodic measure of maximal entropy when \(\lambda_2 = \lambda_1\); the authors however provide a list of examples of maps satisfying this equality.) The proofs depend on the valuative technique developed by the authors, providing efficient tools for controlling the action of polynomial automorphisms on the set of divisors at infinity of all the compactifications of \(\mathbb{C}^2\). To get the complete results is hard work requiring taking care of several technical and delicate details; but the authors did a good job of explaining the arguments they use in a way as clear as possible, and the effort required for reading the paper is well compensated by the results.
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