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Dynamics of rational mappings of multiprojective spaces. (Dynamique des applications rationnelles des espaces multiprojectifs.) (French) Zbl 1046.37026
A rational mapping \(f:\mathbb C^k \to \mathbb C^k\) can be extended to the projective space \(\mathbb P^k \to \mathbb P^k\), where \(\mathbb P^k\) is given by homogeneous coordinates \([z_0:z_1:\dots:z_k]\). The dynamics of \(f\) under iteration is understood best when \(f\) is algebraically stable on \(\mathbb P^k\), i.e., \(\text{deg}(f^j)=(\text{deg}f)^j\).
In the paper under review, \(f\) is considered on a suitable multiprojective space \(\mathbb P^{n_1} \times\dots\times \mathbb P^{n_s}\). The Russakovskii-Shiffman measure \(\mu_f\), which is describing the distribution of preimages of any point, is constructed and discussed by employing a pluripositive Green current. Applications include polynomial skew products and birational polynomial mappings. For a class \({\mathcal H}\) of interesting mappings in \(\mathbb C^2\), it is shown that \(f \in {\mathcal H}\) is algebraically stable in \(\mathbb P^2\) or in \(\mathbb P^1 \times \mathbb P^1\).
Reviewer: Wolf Jung (Aachen)

MSC:
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
31C10 Pluriharmonic and plurisubharmonic functions
32U40 Currents
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
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