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Invariant currents and dynamical Lelong numbers. (English) Zbl 1080.37050

Let \(f=(P_1,\dots,P_k)\) be a polynomial automorphism of \(\mathbb{C}^k\), \(\lambda= \max_j\deg P_j\geq 2\), whose meromorphic extention to \(\mathbb{P}^k=\mathbb{C}^k\cup(t=0)\) maps the hyperplane \((t=0)\) to a single point \(X^+\) lying outside the indeterminacy locus \(I^+\) of \(f\). By results of V. Sibony [in: D. Cerveau et al., Dynamique et géométric complexes, Paris: Société Mathématique de France, Panor. Synth. 8, 97–185 (1999; Zbl 1020.37026)] and N. Sibony and the second author [Ark. Mat. 40, 207–243 (2002; Zbl 1034.37025)], the sequence \(\lambda^{-n} (f^n)^*\omega\) converges to a positive closed current \(T_+\) (Green current of \(f)\) of bidegree \((1,1)\) such that \(f^*T_+=\lambda T_+\) (here \(\omega\) denotes the Fubini-Study form on \(\mathbb{P}^k)\); if \(\lambda> \lambda_2(f):= \lim_{n\to\infty}[\delta_2(f^n)]^{1/n}\), where \(\delta_2(f^n)\) is the degree of \(f^{-n}(L)\) for a generic linear subspace \(L\) of codimension 2, then there exists also a positive closed current \(\sigma_-\) of bidegree \((k-1,k-1)\) and of unit mass such that \((f^{-1})^*\sigma_-=\lambda\sigma_-\).
The authors introduce the dynamical Lelong numbers of positive closed currents \(S\) of bidegree \((1,1)\) and unit mass in \(\mathbb{P}^k\) as \(\partial(S,\sigma_-)=S\wedge \sigma_-(\{X^+\})\). The main result of the paper, Theorem 1.1, states that if \(I^+\) is an attracting set for \(f^{-1}\), then \(\lambda^{-n} (f^n)^+S\to\nu(S, \sigma_-)[t=0]+(1-\nu(S,\sigma_-)]T_+\); moreover, \(\nu(S,\sigma_-)>0\) if and only if the (classical) Lelong number \(\nu(S, X^+)>0\). The result is new even in the case when \(f\) is a complex Hénon mapping.

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
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