Holomorphic maps from 𝐶ⁿ to 𝐶ⁿ

Abstract

We study holomorphic mappings from C n {{\mathbf {C}}^n} to C n {{\mathbf {C}}^n} , and especially their action on countable sets. Several classes of countable sets are considered. Some new examples of Fatou-Bieberbach maps are given, and a nondegenerate map is constructed so that the volume of the image of C n {{\mathbf {C}}^n} is finite. An Appendix is devoted to the question of linearization of contractions.

Proper holomorphic discs avoiding closed convex sets

“…It is enough to prove that there is a proper holomorphic embedding ϕ = (ϕ 1 , ϕ 2 ) : → C | | 2 such that ϕ 2 (ζ) = 0 (ζ ∈ ). By [RR,p. 78] there is a biholomorphic map…”

Universal entire functions for affine endomorphisms of CN

“…2 (2) There are other popular automorphisms of C N which are not affine. For example there are the "shears" introduced by Rosay and Rudin [22], and defined by…”

“…Nevertheless, a complete description of Aut(C N ) (see, for instance, [4,5,22] for a study of some subfamilies of it) is unknown up to date. Although there are plenty of automorphisms of C N , the simplest among them are with no doubt the affine linear mappings (or 'affine endomorphisms')…”

Invariant escaping Fatou components with two rank-one limit functions for automorphisms of

“…It is natural to classify invariant Fatou components both from the point of view of a dynamical characterization (that is, to which model map the iterates are conjugate to) and from the point of view of a geometric characterization (that is, to which model manifold the Fatou component is biholomorphic). The first characterization strongly influences the latter; for example, for polynomial automorphisms of , any invariant Fatou component on which the iterates converge to a fixed point is biholomorphic to [PVW08, RR88, Ued86]. The dynamical characterization is also very related to which types of limit functions there can be in the Fatou component, for example, their rank, and whether the limit sets are in the boundary of the Fatou component or in its interior.…”