Related papers: Introduction to Fatou components in holomorphic dy…
We show that there exist polynomial endomorphisms of C^2, possessing a wandering Fatou component. These mappings are polynomial skew-products, and can be chosen to extend holomorphically of P^2(C). We also find real examples with wandering…
This note initiates the study of the Fatou\,--\,Julia sets of a complex harmonic mapping. Along with some fundamental properties of the Fatou and the Julia sets, we observe some contrasting behaviour of these sets as those with in case of a…
We investigate the existence of wandering Fatou components for polynomial skew-products in two complex variables. In 2004 the non-existence of wandering domains near a super-attracting invariant fiber was shown in [8]. In 2014 it was shown…
We investigate the description of Fatou components for polynomial skew-products in two complex variables. The non-existence of wandering domains near a super-attracting invariant fiber was shown in [L], and the geometrically-attracting case…
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…
Wandering Fatou components were recently constructed by Astorg et al for higher-dimensional holomorphic maps on projective spaces. Their examples are polynomial skew products with a parabolic invariant line. In this paper, we study this…
In this note we review a selection of contemporary research themes in holomorphic dynamics. The main topics that will be discussed are: geometric (laminar and woven) currents and their applications, bifurcation theory in one and several…
We construct polynomial automorphisms with wandering Fatou components. The four-dimensional automorphisms $H$ lie in a one-parameter family, depending on the parameter $\delta \in \mathbb C \setminus \{0\}$, and as $\delta \rightarrow 0$…
The goal of this paper is to study some basic properties of the Fatou and Julia sets for a family of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge 2$. We are particularly interested in studying these sets for semigroups generated by…
For a family of holomorphic functions on an arbitrary domain, we introduce Fatou and Julia like sets, and establish some of their interesting properties.
We investigate the tautness of invariant Fatou components for holomorphic endomorphisms of P^2. Previously, only basins of attraction were known to be taut. We show that two other kinds of recurrent Fatou components are taut. In the first…
This paper studies the geometry of Fatou components in non-Archimedean dynamics. By explicitly computing a wandering domain constructed by Benedetto, it provides the first example of a Fatou component that is an irrational disk.
We prove that the Euclidean ball can be realized as a Fatou component of a holomorphic automorphism of $\mathbb{C}^m$, in particular as the escaping and the oscillating wandering domain. Moreover, the same is true for a large class of…
We study the dynamics of a generic endomorphism $f$ of an Oka-Stein manifold $X$. Such manifolds include all connected linear algebraic groups and, more generally, all Stein homogeneous spaces of complex Lie groups. We give several…
We study invariant Fatou components for holomorphic endomorphisms in $\mathbb{P}^2$. 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…
We show a partial generalization of Sullivan's non-wandering domain theorem in complex dimension two. More precisely, we show the non-existence of wandering Fatou components for polynomial skew products of $ \mathbb{C}^2$ with an invariant…
We study the dynamics of polynomials with coefficients in a non-Archimedean field $K,$ where $K$ is a field containing a dense subset of algebraic elements over a discrete valued field $k.$ We prove that every wandering Fatou component is…
While the dynamics of transcendental entire functions in periodic Fatou components and in multiply connected wandering domains are well understood, the dynamics in simply connected wandering domains have so far eluded classification. We…
We prove the existence of a locally dense set of real polynomial automorphisms of C 2 displaying a wandering Fatou component; in particular this solves the problem of their existence, reported by Bedford and Smillie in 1991. These Fatou…
We completely characterise the bounded sets that arise as components of the Fatou and Julia sets of meromorphic functions. On the one hand, we prove that a bounded domain is a Fatou component of some meromorphic function if and only if it…