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The intersection of laminar currents. (Sur l’intersection des courants laminaires.) (French) Zbl 1048.32021
Let $$\Omega$$ be an open subset of $$\mathbb {D}^2$$, the unit bidisc of $$\mathbb {C}^2$$. A positive $$(1,1)$$ current $$T$$ in $$\Omega$$ is uniformly laminar if there exists a finite measure $$\nu$$ on $$\{ 0 \} \times \mathbb {D}$$ and a family of disjoint graphs $$\Gamma_a$$ in $$\mathbb {D}^2$$ such that $$(0,a) \in \Gamma_a$$ and $$T = \int_{ \{ 0 \} \times \mathbb {D} } [\Gamma_a \cap \Omega ] \,d \nu(a)$$ ($$[J]$$ is the current of integration over $$J$$). A current $$T$$ in $$\Omega$$ is laminar if $$T$$ is an increasing limit of uniformly laminar currents in $$\Omega$$.
The author studies the relationship between the wedge product $$T_1 \wedge T_2$$ of two laminar currents and the measure $$\mu$$ obtained by intersecting the disks filling up $$T_1$$ and $$T_2$$. The intersection is “geometric” if $$\mu =T _1 \wedge T_2$$. The main result is the following. If $$T_1$$ and $$T_2$$ are strongly approximable (SA) currents in $$\Omega$$ with continuous potential, then the product is geometric.
The SA currents is a subclass of laminar currents: $$T$$ is SA if there exists a sequence $$(C_n)_n$$ of analytic subsets in $$\Omega$$ such that $$T$$ is the limit of $${1 \over d_n} [C_n]$$, where $$d_n$$ is the area of $$C_n$$. For instance, the stable and unstable currents of Hénon maps [E. Bedford, M. Lyubich and J. Smillie, Invent. Math. 112, No. 1, 77–125 (1993; Zbl 0792.58034)] and automorphisms of $$K3$$ surface [S. Cantat, Acta Math. 187, No. 1, 1–57 (2001; Zbl 1045.37007)] are SA. Note that the preceding result was already proved in these special cases. Observe also that there exist closed laminar currents with continuous potential whose auto-intersection is not geometric (e.g. the current in $$\mathbb {C}^2$$ given by $$dd^c \max ( \log^+ | z| , \log^+ | w| )$$).

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