Value distribution for sequences of rational mappings and complex dynamics. (English) Zbl 0901.58023

Russakovskii, Alexander; Shiffman, Bernard

Indiana Univ. Math. J. 46, No. 3, 897-932 (1997).

We study pre-images under the iterates \(P^k\) of a rational (not necessarily holomorphic) mapping \(P\) of \(\mathbb{P}^n\). We show, assuming a condition on the topological degree \(\lambda\) of \(P\), that there is a probability measure \(\mu\) on \(\mathbb{P}^n\) and a pluripolar set \({\mathcal E} \subset \mathbb{P}^n\) such that \(\lambda^{-k} P^{k*} \nu\to \mu\) for all probability measures \(\nu\) on \(\mathbb{P}^n \setminus {\mathcal E}\). We also obtain results on the asymptotic equidistribution of the preimages of linear subspaces for sequences of rational mappings between projective spaces.

Reviewer: Bernard Shiffman (Baltimore)

MSC:

37B99	Topological dynamics
30D05	Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F99	Dynamical systems over complex numbers

Keywords:

complex dynamics; rational map; equidistribution; proximity function; plurisubharmonic function; positive current; Green function; Levine current; Fatou set; iteration; probability measure