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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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