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Eigenvaluations. (English) Zbl 1135.37018

This interesting paper studies in a parallel way the dynamics of superattracting holomorphic germs \(f\) at the origin and of polynomial maps \(F\) near infinity, in two complex variables. The problem inspiring this work concerns the dynamical degree: in which sense the dynamical degree measures the rate of attraction of orbits to the origin, respectively infinity? The dynamical degree \(d_\infty\) of a polynomial map \(F\) is defined as the limit of \(\deg(F^n)^{1/n}\), where \(F^n\) is the \(n\)th iterate of \(F\) and \(\deg(F)\) is the algebraic degree of \(F\); analogously, the dynamical degree \(c_\infty\) of a superattracting germ \(f\) is defined as the limit of \(c(f^n)^{1/n}\), where \(c(f)\) is the degree of the first nonvanishing term in the series expansion of \(f\). To avoid trivialities, the authors assume that all the maps and germs involved are dominant, that is their Jacobian determinant does not vanish identically; moreover, they are interested in the superattracting behavior, and so they assume \(c_\infty\), \(d_\infty>1\).
The main result of the paper in the local case is the following: if \(f\) is a dominant superattracting (i.e., \(c_\infty>1\)) holomorphic germ at the origin in \({\mathbb C}^2\), then there exists a plurisubharmonic function \(u\) defined near the origin and with a logarithmic pole at the origin such that \(c_\infty^{-n} u\circ f^n\) decreases to a plurisubharmonic function \(u_\infty\not\equiv-\infty\) with \(u_\infty\circ f= c_\infty u_\infty\). Furthermore, \(c_\infty\) is a quadratic integer, and there exists \(\delta\in(0,1]\) such that \(\delta c_\infty^n\leq c(f^n)\leq c_\infty^n\) for all \(n\geq 1\).
Analogously, the main result in the polynomial case is the following: if \(F\) is a dominant polynomial map with \(d_\infty>1\) in \({\mathbb C}^2\), not conjugate to a skew-product, then there exists a plurisubharmonic function \(U\) with a logarithmic pole at infinity such that \(d_\infty^{-n} U\circ F^n\) decreases to a plurisubharmonic function \(U_\infty\not\equiv-\infty\) with \(U_\infty\circ f= d_\infty U_\infty\). Furthermore, \(d_\infty\) is a quadratic integer, and there exists \(D\in[1,+\infty)\) such that \(d_\infty^n\leq \deg(F^n)\leq D d_\infty^n\) for all \(n\geq 1\).
Possibly even more interesting than the actual results is the method of proof used by the author. They heavily rely on the valuative tree introduced in [C. Favre, M. Jonsson, The valuative tree, Lecture Notes in Mathematics 1853, Berlin: Springer (2004; Zbl 1064.14024)] and already used in other papers by the authors. Roughly speaking, the valuative tree yields an effective way of dealing with all possible sequences of blow-ups of the origin or infinity, replacing the geometric study of the total space of the blow-ups by the algebraic study of valuations on the ring of formal power series in the local case, or on the ring of polynomials in the polynomial case; and these valuations form a tree, endowed with several useful structures. In this paper, the authors show that the germ \(f\) (or the polynomial \(F\)) induces a self-map of the valuative tree with at least one fixed point; and the study of the dynamics near the fixed point on the valuative tree, when translated back in terms of the geometry of a suitable sequence of blow-ups, yields several deep informations about the dynamics of the original map.
For instance, using this technique the authors are able to prove a rigidification result, instrumental in the proofs of the main results of this paper but quite interesting by itself. A holomorphic germ is called rigid if its critical set is contained in a totally invariant set with normal crossing singularities. Rigid germs were classified by the first author [J. Math. Pures Appl., IX. Sér. 79, 475–514 (2000; Zbl 0983.32023)], have simple normal forms, and stay rigid under blow-ups. The authors are able to prove that for any dominant germ \(f\) with \(c_\infty>1\) there is a modification (i.e., a sequence of blow-ups) \(\pi:X\to({\mathbb C}^2,O)\) and a point \(p\in\pi^{-1}(O)\) such that the lift \(\widehat f=\pi^{-1}\circ f\circ\pi\) is holomorphic at \(p\), \(\widehat f(p)=p\) and the germ \(\widehat f:(X,p)\to(X,p)\) is rigid. A similar statement holds for polynomial maps. Since rigid germs are classified (and actually only some of them can appear in this setting) the authors can then exploit the explicit normal forms of rigid germs to deduce their main results.
Reviewer: Marco Abate (Pisa)

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
12J20 General valuation theory for fields
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