##
**Distribution of the values of meromorphic transformations and applications.
(Distribution des valeurs de transformations méromorphes et applications.)**
*(English)*
Zbl 1094.32005

Summary: The meromorphic transform (MT) between compact Kähler manifolds is a surjective multivalued map with an analytic graph. Let \(F_n:X\to X_n\) be a sequence of MT. Let \(\sigma_n\) be an appropriate probability measure on \(X_n\) and \(\sigma\) the product measure of \(\sigma_n\), on \({\mathbf X}= \prod_{n\geq 1}X_n\). We give conditions which imply that
\[
\frac{1}{d(F_n)} [(F_n)^*(\delta_{x_n})- (F_n)^*(\delta_{x_n'})]\to 0
\]
for \(\sigma\)-almost every \({\mathbf x}= (x_1,x_2,\dots)\) and \({\mathbf x}'= (x_1',x_2',\dots)\) in \({\mathbf X}\). Here \(\delta_{x_n}\) in the Dirac mass at \(x_n\) and \(d(F_n)\) the intermediate degree of maximal order of \(F_n\).

We introduce a calculus on MT: intermediate degrees of composition and of product of MT. Using this formalism and what we call the dd\(^c\)-method, we obtain results on the distribution of common zeros, for random \(l\) holomorphic sections of high powers \(L^n\) of a positive holomorphic line bundle \(L\) over a projective manifold.

We also construct the equilibrium measure for random iteration of correspondences. In particular, when \(f:X\to X\) is a meromorphic correspondence of large topological degree \(d_t\), we show that \(d_t^{-n}(f^n)^* \omega^k\) converges to a measure \(\mu\), satisfying \(f^*\mu= d_t\mu\). Moreover, quasi-psh functions are \(\mu\)-integrable. Every projective manifold admits such correspondences. When \(f\) is a meromorphic map, \(\mu\) is exponentially mixing with a precise speed depending on the regularity of the observables.

We introduce a calculus on MT: intermediate degrees of composition and of product of MT. Using this formalism and what we call the dd\(^c\)-method, we obtain results on the distribution of common zeros, for random \(l\) holomorphic sections of high powers \(L^n\) of a positive holomorphic line bundle \(L\) over a projective manifold.

We also construct the equilibrium measure for random iteration of correspondences. In particular, when \(f:X\to X\) is a meromorphic correspondence of large topological degree \(d_t\), we show that \(d_t^{-n}(f^n)^* \omega^k\) converges to a measure \(\mu\), satisfying \(f^*\mu= d_t\mu\). Moreover, quasi-psh functions are \(\mu\)-integrable. Every projective manifold admits such correspondences. When \(f\) is a meromorphic map, \(\mu\) is exponentially mixing with a precise speed depending on the regularity of the observables.

### MSC:

32H50 | Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |

37A50 | Dynamical systems and their relations with probability theory and stochastic processes |