Existence of wandering and periodic domain in given angular region

2020 ◽  
Vol 70 (4) ◽  
pp. 839-848
Author(s):  
Vishnu Narayan Mishra ◽  
Garima Tomar
Keyword(s):  
Entire Functions ◽  
Periodic Domain ◽  
Angular Region ◽  
Wandering Domain ◽  
Structures And Properties ◽  
Fatou Components

AbstractDynamics of composition of entire functions is well related to it's factors, as it is known that for entire functions f and g, fog has wandering domain if and only if gof has wandering domain. However the Fatou components may have different structures and properties. In this paper we have shown the existence of domains with all possibilities of wandering and periodic in given angular region θ.

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 Entire functions for which the escaping set is a spider's web

2011 ◽  
Vol 151 (3) ◽  
pp. 551-571 ◽  
Author(s):  
D. J. SIXSMITH
Keyword(s):  
Entire Functions ◽  
Degree Of Stability ◽  
Transcendental Entire Functions ◽  
Fatou Components

AbstractWe construct several new classes of transcendental entire functions, f, such that both the escaping set, I(f), and the fast escaping set, A(f), have a structure known as a spider's web. We show that some of these classes have a degree of stability under changes in the function. We show that new examples of functions for which I(f) and A(f) are spiders' webs can be constructed by composition, by differentiation, and by integration of existing examples. We use a property of spiders' webs to give new results concerning functions with no unbounded Fatou components.

scholarly journals Bounded Fatou Components of Composite Transcendental Entire Functions with Gaps

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Cunji Yang ◽  
Shaoming Wang
Keyword(s):  
Entire Functions ◽  
Fatou Component ◽  
Transcendental Entire Functions ◽  
Fatou Components

We prove that composite transcendental entire functions with certain gaps have no unbounded Fatou component.

scholarly journals Some entire functions with multiply-connected wandering domains

1985 ◽  
Vol 5 (2) ◽  
pp. 163-169 ◽  
Author(s):  
I. N. Baker
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Julia Set ◽  
Multiply Connected ◽  
Wandering Domain ◽  
Wandering Domains

AbstractA component U of the complement of the Julia set of an entire function ƒ is a wandering domain if the sets ƒn(U) are mutually disjoint, where n ∈ℕ and ƒn is the n-th iterate of ƒ. Examples are given of entire ƒ of order , which have multiply-connected wandering domains. An example is given where the connectivity is infinite.

scholarly journals Orbifold expansion and entire functions with bounded Fatou components

2021 ◽  
pp. 1-40
Author(s):  
LETICIA PARDO-SIMÓN
Keyword(s):  
General Class ◽  
Entire Functions ◽  
Julia Set ◽  
Holomorphic Dynamics ◽  
Fatou Set ◽  
Local Connectivity ◽  
Hyperbolic Functions ◽  
Hyperbolic Orbifold ◽  
Transcendental Entire Functions ◽  
Fatou Components

Abstract Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are functions for which the postsingular set is a compact subset of the Fatou set. Equivalently, they are characterized as being expanding. Mihaljević-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalize these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prove that these maps are also expanding with respect to a suitable orbifold metric, and use this expansion to draw conclusions on the topology and dynamics of the maps. In particular, we generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets. As part of this study, we develop some novel results on hyperbolic orbifold metrics. These are of independent interest, and may have future applications in holomorphic dynamics.

On the Dynamics of Composition of Transcendental Entire Functions in Angular Region-II

2017 ◽  
Vol 84 (3-4) ◽  
pp. 287
Author(s):  
Garima Tomar
Keyword(s):  
Approximation Theory ◽  
Entire Functions ◽  
Angular Region ◽  
Transcendental Entire Functions ◽  
Region Ii

In [14] we showed that for transcendental entire functions f and g, there exist finitely many domains in an angular region, which lie in wandering component of f, wandering component of g and also in wandering component of f ° g and in wandering component of g ° f. Several other related results were discussed in that paper. In this paper we show the existence of such infinite components in angular region, using approximation theory.

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