The intersection of laminar currents.
(Sur l’intersection des courants laminaires.)

*(French)*Zbl 1048.32021Let \(\Omega\) 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 \(dd^c \max ( \log^+ | z| , \log^+ | w| )\)).

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 \(dd^c \max ( \log^+ | z| , \log^+ | w| )\)).

Reviewer: Christophe Dupont (Orsay)

##### 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 |