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Abstract. Let D be a bounded domain in C 2 with a non-compact group of holomorphic automorphisms. Model domains for D are obtained under the hypotheses that at least one orbit accumulates at a boundary point near which the boundary is smooth, real analytic and of finite type.

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Abstract. We consider proper holomorphic mappings of equidimensional pseudoconvex domains in complex Euclidean space, where both source and target can be represented as Cartesian products of smoothly bounded domains. It is shown that such mappings extend smoothly up to the closures of the domains, provided each factor of the source satisfies Condition R. It also shown that the number of smoothly bounded factors in the source and target must be the same, and the proper holomorphic map splits as product of proper mappings between the factor domains.

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.

Let 1 and 2 be strongly pseudoconvex domains in C n and f : 1 → 2 an isometry for the Kobayashi or Carathéodory metrics. Suppose that f extends as a C 1 map to¯ 1 . We then prove that f | ∂ 1 : ∂ 1 → ∂ 2 is a CR or anti-CR diffeomorphism. It follows that 1 and 2 must be biholomorphic or anti-biholomorphic.

For a domain D ⊂ C n , the relationship between the squeezing function and the Fridman invariants is clarified. Furthermore, localization properties of these functions are obtained. As applications, some known results concerning their boundary behavior are extended.

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.