Dynamical properties of families of holomorphic mappings

Abstract

We study some dynamical properties of skew products of Hénon maps of C 2 \mathbb {C}^2 that are fibered over a compact metric space M M . The problem reduces to understanding the dynamical behavior of the composition of a pseudo-random sequence of Hénon mappings. In analogy with the dynamics of the iterates of a single Hénon map, it is possible to construct fibered Green functions that satisfy suitable invariance properties and the corresponding stable and unstable currents. This analogy is carried forth in two ways: it is shown that the successive pull-backs of a suitable current by the skew Hénon maps converges to a multiple of the fibered stable current and secondly, this convergence result is used to obtain a lower bound on the topological entropy of the skew product in some special cases. The other class of maps that are studied are skew products of holomorphic endomorphisms of P k \mathbb {P}^k that are again fibered over a compact base. We define the fibered Fatou component and show that they are pseudoconvex and Kobayashi hyperbolic.