Dynamical properties of families of holomorphic mappings

Abstract: We study some dynamical properties of skew products of H\'{e}non maps of $\mbb C^2$ that are fibered over a compact metric space $M$. The problem reduces to understanding the dynamical behavior of the composition of a pseudo-random sequence of H\'{e}non mappings. In analogy with the dynamics of the iterates of a single H\'{e}non map, it is possible to construct fibered Green's 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 pullbacks of a suitable current by the skew H\'{e}non 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 $\mbb P^k$ that are again fibered over a compact base. We define the fibered basins of attraction and show that they are pseudoconvex and Kobayashi hyperbolic.