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.

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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.

The goal of this paper is to study some basic properties of the Fatou and Julia sets for a family of holomorphic endomorphisms of C k , k ≥ 2. We are particularly interested in studying these sets for semigroups generated by various classes of holomorphic endomorphisms of C k , k ≥ 2. We prove that if the Julia set of a semigroup G which is generated by endomorphisms of maximal generic rank k in C k contains an isolated point, then G must contain an element that is conjugate to an upper triangular automorphism of C k . This generalizes a theorem of Fornaess-Sibony. Secondly, we define recurrent domains for semigroups and provide a description of such domains under some conditions.

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For a domain in the complex plane, we consider the domain S n ( ) consisting of those n × n complex matrices whose spectrum is contained in . Given a holomorphic self-map of S n ( ) such that (A) = A and the derivative of at A is identity for some A ∈ S n ( ), we investigate when the map would be spectrum-preserving. We prove that if the matrix A is either diagonalizable or non-derogatory then for most domains , is spectrum-preserving on S n ( ). Further, when A is arbitrary, we prove that is spectrum-preserving on a certain analytic subset of S n ( ).