The purpose of this note is to explore further the rigidity properties of Hénon maps from [5]. For instance, we show that if H and F are Hénon maps with the same Green measure (µH = µF ), or the same filled Julia set (KH = KF ), or the same Green function (GH = GF ), then H 2 and F 2 have to commute. This in turn, gives that H and F have the same non-escaping sets. Further we prove that, either of the association of a Hénon map H to its Green measure µH or to its filled Julia set KH or to its Green function GH is locally injective.

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.

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The purpose of this paper is to present several examples of non-autonomous basins of attraction that arise from sequences of automorphisms of C k . In the first part, we prove that the non-autonomous basin of attraction arising from a pair of automorphisms of C 2 of a prescribed form is biholomorphic to C 2 . This, in particular, provides a partial answer to a question raised in [2] in connection with Bedford's Conjecture about uniformizing stable manifolds. In the second part, we describe three examples of Short C k 's with specified properties. First, we show that for k ≥ 3, there exist (k − 1) mutually disjoint Short C k 's in C k . Second, we construct a Short C k , large enough to accommodate a Fatou-Bieberbach domain, that avoids a given algebraic variety of codimension 2. Lastly, we discuss examples of Short C k 's with (piece-wise) smooth boundaries.