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

Dinh, Tien-Cuong; Sibony, Nessim

Comment. Math. Helv. 81, No. 1, 221-258 (2006).

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.

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

Keywords:

meromorphic correspondences; dynamical degrees; speed of mixing; random polynomials; compact Kähler manifolds; equilibrium measure; random iteration of correspondences