Dynamical properties of families of holomorphic mappings

TL;DR

This paper investigates the dynamical behavior of skew products of Hénon maps and holomorphic endomorphisms, establishing convergence of currents, entropy bounds, and properties of basins of attraction in complex dynamics.

Contribution

It introduces fibered Green's functions and stable currents for skew products, extending classical dynamics to pseudo-random compositions and higher dimensions.

Findings

Convergence of pullbacks to fibered stable currents

Lower bounds on topological entropy for skew products

Pseudoconvexity and Kobayashi hyperbolicity of basins

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.