We study invariant Fatou components for holomorphic endomorphisms in P 2 . In the recurrent case these components were classified by Sibony and the second author in 1995. In 2008 Ueda completed this classification by proving that it is not possible for the limit set to be a punctured disk. Recently Lyubich and the third author classified non-recurrent invariant Fatou components, under the additional hypothesis that the limit set is unique. Again all possibilities in this classification were known to occur, except for the punctured disk. Here we show that the punctured disk can indeed occur as the limit set of a non-recurrent Fatou component. We provide many additional examples of holomorphic and polynomial endomorphisms of C 2 with non-recurrent Fatou components on which the orbits converge to the regular part of arbitrary analytic sets.
In this paper we construct for every integer n > 1 a complex manifold of dimension n which is exhausted by an increasing sequence of biholomorphic images of C n (i.e., a long C n ), but it does not admit any nonconstant holomorphic or plurisubharmonic functions (see Theorem 1.1). Furthermore, we introduce new holomorphic invariants of a complex manifold X, the stable core and the strongly stable core, that are based on the long term behavior of hulls of compact sets with respect to an exhaustion of X. We show that every compact polynomially convex set B ⊂ C n such that B =B is the strongly stable core of a long C n ; in particular, holomorphically nonequivalent sets give rise to nonequivalent long C n 's (see Theorems 1.2 and 1.6 (a)). Furthermore, for every open set U ⊂ C n there exists a long C n whose stable core is dense in U (see Theorem 1.6 (b)). It follows that for any n > 1 there is a continuum of pairwise nonequivalent long C n 's with no nonconstant plurisubharmonic functions and no nontrivial holomorphic automorphisms. These results answer several long standing open problems.
We study the geometry of simply connected wandering domains for entire functions and we prove that every bounded connected regular open set, whose closure has a connected complement, is a wandering domain of some entire function. In particular such a domain can be realized as an escaping or an oscillating wandering domain. As a consequence, we obtain that every Jordan curve is the boundary of a wandering Fatou component of some entire function.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.