An area approach to wandering domains for smooth surface endomorphisms

1991 ◽  
Vol 11 (1) ◽  
pp. 181-187 ◽  
Author(s):  
Alec Norton
Keyword(s):  
Smooth Surface ◽  
Main Lemma ◽  
Rational Map ◽  
Wandering Domain ◽  
Direct Corollary ◽  
Simply Connected ◽  
Simply Connected Domains ◽  
Wandering Domains

AbstractWe prove an infinite-area lemma for maps which are area non-contracting on the boundaries of certain domains; the maps are required to be smooth to the extent that their Jacobians are twice differentiable (see Main Lemma below). It will follow that a hyperbolic rational map has no wandering simply connected domains. As a more direct corollary, a C3 diffeomorphism f of a compact smooth 2-manifold cannot have a wandering domain Δ if f is area non-contracting on the boundary of each forward image of Δ.

scholarly journals Escaping Fatou components of transcendental self-maps of the punctured plane

2019 ◽  
pp. 1-25 ◽  
Author(s):  
DAVID MARTÍ-PETE
Keyword(s):  
Approximation Theory ◽  
Entire Functions ◽  
Holomorphic Functions ◽  
Sufficient Condition ◽  
Wandering Domain ◽  
Simply Connected ◽  
Fatou Components ◽  
Wandering Domains

Abstract We study the iteration of transcendental self-maps of $\,\mathbb{C}^*\!:=\mathbb{C}\setminus \{0\}$ , that is, holomorphic functions $f:\mathbb{C}^*\to\mathbb{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0,\infty\}$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\,\mathbb{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a sufficient condition for a function to have a simply connected escaping wandering domain. Finally, we remark that our results also provide new examples of entire functions with escaping Fatou components.

scholarly journals Classifying simply connected wandering domains

2021 ◽  
Author(s):  
Anna Miriam Benini ◽  
Vasiliki Evdoridou ◽  
Núria Fagella ◽  
Philip J. Rippon ◽  
Gwyneth M. Stallard
Keyword(s):  
Entire Functions ◽  
General Technique ◽  
Jordan Curves ◽  
Wandering Domain ◽  
Detailed Classification ◽  
Transcendental Entire Functions ◽  
Simply Connected ◽  
Fatou Components ◽  
Wandering Domains

AbstractWhile 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 give a detailed classification of the dynamics in such wandering domains in terms of the hyperbolic distances between iterates and also in terms of the behaviour of orbits in relation to the boundaries of the wandering domains. In establishing these classifications, we obtain new results of wider interest concerning non-autonomous forward dynamical systems of holomorphic self maps of the unit disk. We also develop a new general technique for constructing examples of bounded, simply connected wandering domains with prescribed internal dynamics, and a criterion to ensure that the resulting boundaries are Jordan curves. Using this technique, based on approximation theory, we show that all of the nine possible types of simply connected wandering domain resulting from our classifications are indeed realizable.

scholarly journals Universal Taylor series for non-simply connected domains

2010 ◽  
Vol 348 (9-10) ◽  
pp. 521-524 ◽  
Author(s):  
Stephen J. Gardiner ◽  
Nikolaos Tsirivas
Keyword(s):  
Taylor Series ◽  
Universal Taylor Series ◽  
Simply Connected ◽  
Simply Connected Domains

Hülder Conditions and the Topology of Simply Connected Domains*

1983 ◽  
Vol 26 (2) ◽  
pp. 189-191
Author(s):  
Dov Aharonov
Keyword(s):  
General Result ◽  
Unit Disc ◽  
Harmonic Functions ◽  
Simply Connected ◽  
Simply Connected Domains

AbstractLet ƒ be regular univalent and normalized in the unit disc U (i.e. ƒ ∊ S) and continuous on U ∈ T, where T denotes the boundary of U.Recently Essén proved [5] a conjecture of Piranian [7] stating that if the derivative of ƒ ∊ S is bounded in U and ƒ(z1) = ƒ(z2) = … = ƒ(zn) for Zj ∊ T, 1 ≤ j ≤ n, then n ≤ 2. In fact, Essén proved a more general result, using a deep result on harmonic functions. The aim of the following article is to replace Essén's proof by a completely different proof which is based only on Goluzin's inequalities and is much more elementary.

Sign in / Sign up

Export Citation Format

Share Document