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Sequences of multivalued meromorphic mappings and laminar currents. (Suites d’applications méromorphes multivaluées et Courants laminaires.) (French) Zbl 1085.37039
Let \((X_1,\omega_1)\) and \((X_2,\omega_2)\) be the compact Kählerian varieties of dimensions \(k_1\) and \(k_2\), respectively. Let \(F_n: X_1\to X_2\) be the meromorphic (multi-valued) mappings. For \(F_n\), denote by \(d_n\) the topological degree and \(\lambda_n\) the intermediate degree of the order \(k_2-1\). Suppose that the series \(\sum \lambda_n d_n^{-1}\) converges. For the case of the rational mappings between projective spaces, Sodin-Russakovskii-Shiffman showed that the preimages of \(F_n\) are equidistributed when \(n\) tends to infinity. More precisely, \(d_n^{-1}F_n^\ast(\delta_z)-d_n^{-1}F_n^\ast(\delta_{z'})\) tends weakly to 0 for \(z, z'\in X_2\) out of a pluripolar set \(\mathcal E\), where \(\delta_z\) denotes the Dirac mass in \(z\). If \(\{F_n\}=\{f^n\}\) is the sequence of the iterations of a meromorphic (multivalued) mapping \(f\) whose topological degree is bigger than the other dynamic degrees, \(\mathcal E\) is contained in a countable union of analytic ensembles. Besides, \(d_n^{-1}F_n^\ast(\delta_z)\) converges to the equilibrium measure \(\mu\) of \(f\). This measure also reflects the distribution of the repelling fixed-points of \(F_n\).
The author looks for the more general conditions on the sequence \(\{F_n\}\) in order that \(\mathcal E\) be contained in a countable union of analytic ensembles. It is shown that it is the case when the sequence of postcritical currents \(S_n\) of \(F_n\) converges to a current \(S_\infty\). Then the ensemble \(\mathcal E\) is contained essentially in \(\{\nu(S_\infty, z)> 0\}\), where \(\nu(S_\infty, z)\) denotes the Lelong number of \(S_\infty\) in \(z\). The distribution of the repelling fixed-points in the case when \(X_1\) and \(X_2\) are a compact Riemann surface \(X\) is also studied. Suppose that for all \(z\notin\mathcal E\), \(d_n^{-1}F_n^\ast(\delta_z)\) weakly tends to a measure \(\mu\) that is singular with respect to \(S_\infty\). Then the measure, equidistributed in the repelling stationary fixed-points of \(F_n\), tends weakly to \(\mu\). This result reflects the connection between the postcritical ensembles and the equidistribution of the repelling fixed-points of \(F_n(=f^n)\).
Giving an example, it is shown that the hypothesis of the connection between \(\mu\) and \(S_\infty\) is necessary. The author constructs the local inverse mappings of \(F_n\)-branches, which are defined on small balls and admit, as images, the ensembles whose diameters tend to \(0\) when \(n\to\infty\). This imply the equidistribution of preimages. Then the author studies the laminarity of certain positive closed currents of any dimension. Let \(\{Z_n\}\) be a sequence of images of \(\mathbb P^s\) in a projective variety \(X\) of dimension \(k>s\). Suppose that the sequence of integration currents on \(Z_n\), normalized by the appropriate way, tends to a current \(T\). Then \(T\) is tissued (geometric) and is laminar if \(s=k-1\) and the singularities of \(Z_n\) are sufficient. The construction of inverses branches is used to prove this result. More precisely, it is considered the projections \(F_n\) of \(Z_n\) on a projective space \(\mathbb P^s\). The inverse \(F_n\)-branches of the open sets of \(\mathbb P^s\), with controlled size, form the normal families of complex varieties in \(X\). Passing to the limit, one gets the complex varieties that form the current \(T\).
As an application, it is shown that the Green currents of some bidimensions, of a regular polynomial automorphism, are either laminar or tissued. The laminarity of Green currents is also valid for the automorphisms of a projective variety. To this end the author studies the laminarity of the limits \(T\) of \(\{Z_n\}\) when these analytic ensembles are not the images of \(\mathbb P^s\). Here it is used a notion of dual variety (curvature variety) \({\widehat Z}_n\) that is well adapted to the dynamic problems. In a certain sense it permits to work with the derivatives in the form of geometric objects. When the volume of \({\widehat Z}_n\) verifies vol\(({\widehat Z}_n) = O(\text{vol}(Z_n))\), then the current limits \(T\) are also tissued.

37F05 Dynamical systems involving relations and correspondences in one complex variable
32C30 Integration on analytic sets and spaces, currents
32V40 Real submanifolds in complex manifolds
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
32H04 Meromorphic mappings in several complex variables
32U40 Currents
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