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Superattractive fixed points in \({\mathbb{C}}^ n\). (English) Zbl 0858.32023
A fixed point \(x_0\) of an analytic mapping \(f\) is called superattractive if there is a neighborhood \(U\) of \(x_0\) such that \(|f^{0n} (y)-x_0 |\leq C |f^{0(n-1)} (y)-x_0 |^2\) for some constant \(C\) and all \(n>0\), \(y\in U\). In one dimension, such an \(f\) is analytically conjugate to an exponential map; that is, there exists \(\varphi\) analytic near \(x_0\) with \(\varphi(f(x)) = \varphi (x)^k\) for some \(k\geq 2\). In higher dimensions, the orbit of the critical locus of \(f\) near \(x_0\) must be preserved under any conjugacy, even topological. For homogeneous maps, this orbit will consist of smooth curves, while for many nonhomogeneous maps, this orbit will contain curves with singularities at \(x_0\). Hence there can be no conjugacy in general to the lowest-order terms of \(f\). The goal of this paper is to use potential-theoretical methods to relate the lowest order terms of \(f\) to the dynamics near \(x_0\).
The authors consider the case of an analytic mapping \(F:U\to \mathbb{C}^{n+1}\), where \(U\) is a neighborhood of 0 and \(F(x)=F_k(x) + F_{k+1} (x) + \cdots\), with each \(F_j\) homogeneous of degree \(j\) and satisfying \(k\geq 2\) and \(F_k(x) = 0\) if and only if \(x=0\). They first introduce the plurisubharmonic Green function \(h_F\) for \(F\), develop some general results on currents, and then turn to the homogeneous case \(F=F_k\). In this case, \(F\) induces an analytic map \(f\) on \(\mathbb{P}^n\). In the case \(n=1\), the authors provide a simple proof of the existence of the Brolin measure \(\mu_f\) on \(\mathbb{P}^1\) and relate it to \(h_F\) by the formula \((1/2 \pi) dd^ch_F = \pi^*\mu_f\), where \(\pi: \mathbb{C}^2 \backslash \{0\}\to \mathbb{P}^1\) is the canonical projection. They then generalize this proof to higher dimensions to get an invariant (1,1)-current \(\omega_f\) with the same relation to \(h_F\) and give an example for which \(h_F\) and \(\omega_f\) are computable.
The authors then study the basin of attraction \(\Omega_F\) of 0 for \(F\), which is pseudoconvex, and use \(\omega_f\) to define the “Levi-current” of \(\partial \Omega_F\), which is an analogue of the Levi form. This interpretation provides a foliation of the part of \(\partial \Omega_F\) which does not project onto the Julia set of \(f\). This is developed in some detail for the case \(n=1\), \(f\) a polynomial.
In the last section, the authors turn to the nonhomogeneous case and use the results developed earlier to give a limited form of conjugacy between two maps \(F\) and \(\widetilde F\) whose lowest-order terms agree.

32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
32C30 Integration on analytic sets and spaces, currents
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37B99 Topological dynamics
31C10 Pluriharmonic and plurisubharmonic functions
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