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The dynamics of semigroups of transcendental entire functions II

Authors:
Dinesh Kumar
Dinesh Kumar
Dinesh Kumar
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Sanjay Kumar at Deen Dayal Upadhyaya College (University of Delhi)
Sanjay Kumar
  • Deen Dayal Upadhyaya College (University of Delhi)
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Abstract

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.
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... Later in 1998, Poon [10] studied the Fatou and Julia sets of holomorphic semigroup generated by transcendental entire functions. Due to them [4,10], the Fatou and Julia set of holomorphic semigroup H are denoted by F (H) and J(H) and defined by The escaping set of holmorphic semigroup was firstly studied by Kumar and Kumar [6] in 2016. The escaping set of holomorphic semigroup H due to Kumar and Kumar [6] is defined as ...
... Due to them [4,10], the Fatou and Julia set of holomorphic semigroup H are denoted by F (H) and J(H) and defined by The escaping set of holmorphic semigroup was firstly studied by Kumar and Kumar [6] in 2016. The escaping set of holomorphic semigroup H due to Kumar and Kumar [6] is defined as ...
... They [6] proved that I (H) may be empty and I (H) ⊂ h∈H I(h). Also, they [6] proved that for any transcendental semigroup, it is forward invariant and later Kumar et.al. ...
Escaping set and Bounded Orbit Set of Holomorphic Semigroup on C
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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.
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... For recent developments in Fatou Julia theory the reader is urged to refer to [1,2,4,19,26]. A natural generalization of the Fatou and Julia theory is the dynamics of semigroups of meromorphic functions initiated by A. Hinkkanen and G. Martin (see [13,14]) for rational functions and for transcendental entire functions this study is initiated by K.K. Poon (see [23,24], also see [9,13,14,21] ). A semigroup F of entire functions is a semigroup with binary operation defined by the function composition. ...
... The investigations initiated in [5] and the present paper are just the initial stages and there are many aspects to be looked into. For example, one can look into the domains like the wandering domains and Baker's domains of transcendental semigroups ( see [9,15]), and quasi-nested wandering domains(see [12] and [20]). The interesting point is that this study differs from dynamics of a meromorphic function as well as the dynamics of semigroups in the sense that many properties of the dynamics of meromorphic function and that of dynamics of transcendental semigroups fail to hold in the present Fatou and Julia like theory. ...
Fatou and Julia like sets II
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  • Dec 2021
This paper is a continuation of authors work: Fatou and Julia like sets, Ukranian Math. J., to appear/arXiv:2006.08308[math.CV](see [5]). Here, we introduce escaping like set and generalized escaping like set for a family of holomorphic functions on an arbitrary domain, and establish some distinctive properties of these sets. The connectedness of the Julia like set is also proved.
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... For recent developments in Fatou Julia theory the reader is urged to refer to [1,2,4,19,26]. A natural generalization of the Fatou and Julia theory is the dynamics of semigroups of meromorphic functions initiated by A. Hinkkanen and G. Martin (see [13,14]) for rational functions and for transcendental entire functions this study is initiated by K.K. Poon (see [23,24], also see [9,13,14,21] ). A semigroup F of entire functions is a semigroup with binary operation defined by the function composition. ...
... The investigations initiated in [5] and the present paper are just the initial stages and there are many aspects to be looked into. For example, one can look into the domains like the wandering domains and Baker's domains of transcendental semigroups ( see [9,15]), and quasi-nested wandering domains(see [12] and [20]). The interesting point is that this study differs from dynamics of a meromorphic function as well as the dynamics of semigroups in the sense that many properties of the dynamics of meromorphic function and that of dynamics of transcendental semigroups fail to hold in the present Fatou and Julia like theory. ...
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This paper is a continuation of authors work: Fatou and Julia like sets,Ukranian J. Math., to appear/arXiv:2006.08308[math.CV](see [4]). Here, we introduce escaping like set and generalized escaping like set for a family of holomorphic functions on an arbitrary domain, and establish some distinctive properties of these sets. The connectedness of the Julia like set is also proved.
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... The motivation for the proof comes from [13] Let's consider two commuting functions of class B i.e. f • g = g • f . Functions of class B have the property that J(f ) = J(g) and therefore J(f • g) = J(f ) [10], which implies that F (f • g) = F (f ) = F (g). This implies that a point z 0 ∈ J(f • g) can't belong to F (g) (or F (f )), because that would imply that z 0 ∈ F (f • g) as well, and that's a clear contradiction (since I(f • g) is contained in J(f )). ...
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... By Lemma 3.4, we can say that U is unbounded. From [11], we know that if F(G) has an unbounded component then all the components of F(G) is simply connected. So G does not contain any multiply-connected Fatou component and consequently does not have any Baker wandering domain. ...
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... In the second section, we have extended the results of nearly abelian rational semigroups to the best possible class of nearly abelian transcendental semigroups. Moreover, we have extended some of the results of [13,14,15] to nearly abelian transcendental semigroups. ...
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The notion of nearly abelian rational semigroup was introduced by Hinkannen and Martin. In this paper, we have introduced the notion of nearly abelian transcendental semigroup. We have extended the results of nearly abelian rational semigroups to the best possible class of nearly abelian transcendental semigroups. We have given a class of functions which nearly permute with a given transcendental entire function. In addition, a relation between the postsingular set of composition of two entire functions with that of individual functions is obtained.
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... 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. ...
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