# Distribution des valeurs de transformations méromorphes et applications

### Tien-Cuong Dinh

Institut Mathématique de Jussieu, Paris, France### Nessim Sibony

Université Paris-Sud, Orsay, France

## Abstract

A meromorphic transform (MT) between compact Kähler manifolds is a surjective multivalued map with an analytic graph. Let $F_{n}:X→X_{n}$ be a sequence of MT. Let $σ_{n}$ be an appropriate probability measure on $X_{n}$ and $σ$ the product measure of $σ_{n}$, on \( \XX:=\prod_{n\geq 1} X_n \). We give conditions which imply that

for $σ$-almost every \( \xx=(x_1,x_2,\ldots) \) and \( \xx'=(x_1',x_2',\ldots) \) in $X$. Here $δ_{x_{n}}$ is 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 {\it 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 $f:X→X$ is a meromorphic correspondence of large topological degree $d_{t}$, we show that $d_{t}(f_{n})_{∗}ω_{k}$ converges to a measure $μ$, satisfying $f_{∗}μ=d_{t}μ$. Moreover, quasi-psh functions are $μ$-integrable. Every projective manifold admits such correspondences. When $f$ is a meromorphic map, $μ$ is exponentially mixing with a precise speed depending on the regularity of the observables.

## Cite this article

Tien-Cuong Dinh, Nessim Sibony, Distribution des valeurs de transformations méromorphes et applications. Comment. Math. Helv. 81 (2006), no. 1, pp. 221–258

DOI 10.4171/CMH/50