Abstract
The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can occur for polynomials, there also exist so-called Baker domains, periodic components where all orbits converge to infinity, as well as wandering domains. In trying to find analogues of these one dimensional results, it is not clear which higher dimensional transcendental maps to consider. In this paper we find inspiration from the extensive work on the dynamics of complex Hénon maps. We introduce the family of transcendental Hénon maps, and study their dynamics, emphasizing the description of Fatou components. We prove that the classification of the recurrent invariant Fatou components is similar to that of polynomial Hénon maps, and we give examples of Baker domains and wandering domains.
Highlights
We introduce the family of transcendental Hénon maps, and study their dynamics, emphasizing the description of Fatou components
We prove that the classification of the recurrent invariant Fatou components is similar to that of polynomial Hénon maps, and we give examples of Baker domains and wandering domains
Very little is known about the dynamics of holomorphic automorphisms of C2, there have been results showing holomorphic automorphisms of C2 with interesting dynamical behavior, such as the construction of oscillating wandering domains by Sibony and the third named author [20], and a result of Vivas, Wold and the last author [28] showing that a generic volume preserving automorphisms of C2 has a hyperbolic fixed point with a stable manifold which is dense in C2
Summary
Let n ∈ N, n ≥ 1 and let X be a complex manifold. There are (at least) two natural definitions of what it means for a family F ⊂ Hol(X, Cn) to be normal. Definition 2.1 A family F ⊂ Hol(X, Cn) is Pn-normal if for every sequence ( fn) ∈ F there exists a subsequence ( fnk ) converging uniformly on compact subsets to f ∈ Hol(X, Pn). A family F ⊂ Hol(X, Cn) is Cn-normal if for every sequence ( fn) ∈ F which is not divergent on compact subsets there exists a subsequence ( fnk ) converging uniformly on compact subsets to f ∈ Hol(X, Cn) This is equivalent to F being relatively compact in Hol(X, Cn) ∪ ∞ ⊂ C0(X, Cn). If the sequence N j increases sufficiently fast, for 1 < |z0| ≤ 2 we have that Fn(z0, w0) → [0 : 1 : 0] ∈ ∞, again uniformly on compact subsets It follows that D × C is a P2-Fatou component.
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