Dynamical properties of families of holomorphic mappings

Abstract

Abstract. We study some dynamical properties of skew products of Hénon maps of C 2 that are fibered over a compact metric space M . The problem reduces to understanding the dynamical behavior of the composition of a pseudorandom 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 that are again fibered over a compact base. We define the fibered Fatou component and show that they are pseudoconvex and Kobayashi hyperbolic.

Ergodic properties of families of Hénon maps

“…That the support of µ ± Λ is J ± Λ and the correspondence Λ → J ± Λ is lower semi-continuous can be shown as in Proposition 1.2 in [15].…”

“…That G + Λ is a strictly positive pluriharmonic function on C 2 \ K + Λ and vanishes precisely on K + Λ follows by using arguments similar to those in Proposition 1.1 in [15].…”

“…That the correspondence Λ → G + Λ is continuous follows by using arguments similar to those in Proposition 1.1 in [15].…”

“…Let v Λn := u n − q + Λ n+1 . By Proposition 1.1 in [15] (See (2.15) therein), one can show that q + Λn is bounded on U by a fixed constant for all n ≥ 1. This implies that there exists A 1 > 0 such that |v Λn | ≤ A 1 on U for all n ≥ 1.…”

“…This classification has 7 families and several of them are skew products of Hénon maps for suitable choices of the parameters that are involved. It must also be mentioned that Coman-Fornaess [5] have studied the dynamics of these classes of automorphisms by treating them as self-maps of C 3 ; the view point adopted here (and in [15]) is to think of them as skew products with the intention of studying their ergodic properties. In [15], the hypotheses on σ were rather general -in fact, σ was assumed to be only continuous.…”