Content uploaded by Sanjay Kumar
Author content
All content in this area was uploaded by Sanjay Kumar on Jan 22, 2015
Content may be subject to copyright.
A preview of the PDF is not available
Escaping set and Julia set of transcendental semigroups
Abstract
We discuss the dynamics of an arbitrary semigroup of transcendental entire
functions using Fatou-Julia theory and provide some condition for the complete
invariance of escaping set and Julia set of transcendental semigroups. Some
results on limit functions and postsingular set have been discussed. A class of
hyperbolic transcendental semigroups and semigroups having no wandering domains
have also been provided.
... Also, they [6] proved that for any transcendental semigroup, it is forward invariant and later Kumar et.al. [7] proved that for any abelian transcendental semigroup escaping set is backward invariant. Also, they [7] proved that J(H) ∩ I (H) = ∅ and J(H) = ∂I (H) where J(H) is Julia set of transcendental entire semigroup H. Later, Subedi [5] introduced another version of escaping set of holomorphic semigroup. ...
... [7] proved that for any abelian transcendental semigroup escaping set is backward invariant. Also, they [7] proved that J(H) ∩ I (H) = ∅ and J(H) = ∂I (H) where J(H) is Julia set of transcendental entire semigroup H. Later, Subedi [5] introduced another version of escaping set of holomorphic semigroup. Due to Subedi the escaping set of holomorphic semigroup H is defined by ...
In this paper, we study the structure and properties of escaping sets of holomorphic semigroups. In particular, we study the relationship between escaping set of holomorphic semigroup and escaping set of each function that lies in that semigroup. We also study about the invariantness of escaping sets. Also, in this paper, we define the term bounded orbit set K(H) and the set K'(H) of holomorphic semigroup H. Then we study their invariantness and their relations with escaping sets. We also construct a particular class of holomorphic semigroups generated by two holomorphic functions such that bounded orbit set of holomorphic semigroup is equal to bounded orbit set of its generators.
... , J(f ) and I(f ). For the existing results of Fatou Julia theory under transcendental semigroup, we refer [4,5,6,8,9]. ...
In this paper, we mainly study hyperbolic semigroups from which we get non-empty escaping set and Eremenko's conjecture remains valid. We prove that if each generator of bounded type transcendental semigroup S is hyperbolic, then the semigroup is itself hyperbolic and all components of I(S) are unbounded
... Both of them naturally generalized classical complex dynamics to the dynamics of the sequence of different functions by means of composition. Another motivation of studying escaping set of transcendental semigroup comes from the work of Dinesh Kumar and Sanjay Kumar [6,7] where they defined escaping set and discussed how far escaping set of classical transcendental dynamics can be generalized to semigroup dynamics. ...
This is an expository plus research paper which mainly exposes preliminary connection and contrast between classical complex dynamics and semigroup dynamics of holomorphic functions. Classically, we expose some existing results of rational and transcendental dynamics and we see how far these results generalized to holomorphic semigroup dynamics as well as we also see what new phenomena occur.
... Both of them naturally generalized some results of classical complex dynamics to the semigroup dynamics as well as they also investigated some new phenomena in semigroup dynamical system. Another motivation of studying escaping sets and some extra properties and structure of Fatou sets and Julia sets of transcendental semigroups comes from the work of Dinesh Kumar and Sanjay Kumar [8,9,10] where they defined escaping set and discussed how far escaping set of classical transcendental dynamics can be generalized to semigroup dynamics. In parallel, we also studied certain structure and propperties of Fatou sets, Julia sets and escaping sets under semigroup dynamics in [17,18,19,20,21,22,23,24]. From these attempts, we again more motivated to the study of fast escaping sets of transcendental semigroups. ...
... Both of them naturally generalized some results of classical complex dynamics to the semigroup dynamics as well as they also investigated some new phenomena in semigroup dynamical system. Another motivation of studying escaping sets and some extra properties and structure of Fatou sets and Julia sets of transcendental semigroups comes from the work of Dinesh Kumar and Sanjay Kumar [8,9,10] where they defined escaping set and discussed how far escaping set of classical transcendental dynamics can be generalized to semigroup dynamics. In parallel, we also studied certain structure and propperties of Fatou sets, Julia sets and escaping sets under semigroup dynamics in [17,18,19,20,21,22,23,24]. From these attempts, we again more motivated to the study of fast escaping sets of transcendental semigroups. ...
... Both of them naturally generalized classical complex dynamics to the dynamics of the sequence of different functions by means of composition. Another motivation of studying escaping set of transcendental semigroup comes from the work of Dinesh Kumar and Sanjay Kumar [8,9] where they defined escaping set and discussed how far escaping set of classical transcendental dynamics can be generalized to semigroup dynamics. In parallel, we also studied structure and propperties of Fatou set, Julia set and escaping under semigroup dynamics in [16,17,18,19,20,21,22,23]. From these attempts, we again more motivate to study fast escaping set of transcendental semigroup. ...
In this paper, we study fast escaping set of transcendental semigroup. We discuss some the structure and properties of fast escaping set of transcendental semigroup. We also see how far the classical theory of fast escaping set of transcendental entire function applies to general settings of transcendental semigroups and what new phenomena can occur.
Given two permutable entire functions f and g, we establish vital relationship between escaping sets of entire functions f, g and their composition. We provide some families of transcendental entire functions for which Eremenko’s conjecture holds. In addition, we investigate the dynamical properties of the mapping
In this poster presentation, we show under what conditions the Fatou, Julia, and escaping sets of a holomorphic semigroup are respectively equal to the Fatou, Julia and escaping sets of its proper subsemigroups.
We mainly generalize the notion of abelian transcendental semigroup to nearly abelian transcendental semigroup. We prove that Fatou set, Julia set and escaping set of nearly abelian transcendental semigroup are completely invariant. We investigate no wandering domain theorem in such a transcendental semigroup. We also obtain results on a complete generalization of the classification of periodic Fatou components.
We investigate to what extent Fatou set, Julia set and escaping set of transcendental semigroup is respectively equal to the Fatou set, Julia set and escaping set of its subsemigroup. We define partial fundamental set and fundamental set of transcendental semigroup and on the basis of this set, we prove that Fatou set and escaping set of transcendental semigroup S are non-empty.
We study the dynamics of an arbitrary semigroup of transcendental entire
functions using Fatou-Julia theory. Several results of the dynamics associated
with iteration of a transcendental entire function have been extended to
transcendental semigroups. We provide some conditions for connectivity of the
Julia set of the transcendental semigroups. We also study finitely generated
transcendental semigroups, abelian transcendental semigroups and limit
functions of transcendental semigroups on its invariant Fatou components.
In this paper, the dynamics on a transcendental entire semigroup G is investigated. We show the possible values of any limit function of G in strictly wandering domains and Fatou components, respectively. Moreover, if G is of class , for any in a Fatou domain, there does not exist a sequence of G such that as .
We study the dynamics of an arbitrary semigroup of transcendental entire functions using Fatou-Julia theory. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental semigroups. We provide some conditions for connectivity of the Julia set of the transcendental semigroups. We also study finitely generated transcendental semigroups, abelian transcendental semigroups and limit functions of transcendental semigroups on its invariant Fatou components.
We consider dynamics of semigroups of rational functions on Riemann sphere. First, we will define hyperbolic rational semigroups and show the metrical property. We will also define sub-hyperbolic and semi-hyperbolic rational semigroups and show no wandering domain theorems. By using these theorems, we can show the continuity of the Julia set with respect to the perturbation of the generators. By using a method similar to that in (Y), we can show that if a finitely generated rational semigroup is semi-hyperbolic and satisfies the open set condition with the open set O satisfying $\#(\partial O\cap J(G))
Let f be an entire function of finite order whose set of singular values is bounded or, more generally, a finite composition of such functions. We show that every escaping point of f can be connected to ∞ by a curve in I(f). This provides a positive answer to a question of Eremenko for a large class of entire functions.