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.


Introduction
The principal aim of this paper is to study fast escaping set not for iterates of single transcendental entire maps of complex plane C but for the composite of the family F of such maps. Let F be a space of transcendental entire maps on C. For any map f ∈ F , C is naturally partitioned into two subsets: the set of normality and its complement. The set of normality or This conjecture is considered as an important open problem of transcendental dynamics and nowadays it is famous as Eremenko's conjecture. The Eremenko's conjecture 1.1 in general case has been proved by using the fast escaping set A(f ), which consists of points whose iterates tends to infinity as fast as possible. This set is a subset of escaping set and it was introduced first time by Bergweiler and Hinkkanen [3] and defined in the following form by Rippon and Stallard [14]. For a transcendental entire function f , the fast escaping set is a set of the form: A(f ) = {z ∈ C : ∃L ∈ N such that |f n+L (z)| M n (R) for n ∈ N} where M(r) = max |z|=r |f (z)|, r > 0 and M n (r) denotes nth iteration of M(r) with respect to r. R > 0 can be taken any value such that M(r) > r for r R.
Recall that the set C(f ) = {z ∈ C : f ′ (z) = 0} is the set of critical points of the transcendental entire function f and the set CV (f ) = {w ∈ C : w = f (z) such that f ′ (z) = 0} is called the set of critical values. The set AV (f ) consisting of all w ∈ C such that there exists a curve (asymptotic path) Γ : [0, ∞) → C so that Γ(t) → ∞ and f (Γ(t)) → w as t → ∞ is called the set of asymptotic values of f and the set SV (f ) = (CV (f ) ∪ AV (f )) is called the singular values of f . If SV (f ) has only finitely many elements, then f is said to be of finite type. If SV (f ) is a bounded set, then f is said to be of bounded type. The sets S = {f : f is of finite type} and B = {f : f is of bounded type} are respectively called Speiser class and Eremenko-Lyubich class.
The main concern of such a transcendental iteration theory is to describe the nature of the components of Fatou set and the structure and properties of the Julia set, escaping set and fast escaping set. We use monograph: dynamics of transcendental entire functions [7] and book: holomorphic dynamics [10] for basic facts concerning the Fatou set, Julia set and escaping set of a transcendental entire function. We use [13,14,15] for facts and results concerning the fast escaping set of a transcendental entire function.
Our particular interest is to study of the dynamics of the families that are semigroups generated by the class F of transcendental entire maps. For a collection F = {f α } α∈∆ of such maps, let S = f α be a transcendental semigroup generated by them. The index set ∆ to which α belongs is allowed to be infinite in general unless otherwise stated. Here, each f ∈ S is a transcendental entire function and S is closed under functional composition. Thus, f ∈ S is constructed through the composition of finite number of functions f α k , (k = 1, 2, 3, . . . , m). That is, f = f α 1 •f α 2 •f α 3 •· · ·•f αm . A semigroup generated by finitely many transcendental entire functions f i , (i = 1, 2, . . . , n) is called finitely generated transcendental semigroup. We write S = f 1 , f 2 , . . . , f n . If S is generated by only one transcendental entire function f , then S is cyclic transcendental semigroup. We write S = f . In this case, each g ∈ S can be written as g = f n , where f n is the nth iterates of f with itself. Note that in our study of semigroup dynamics, we say S = f a trivial semigroup. The transcendental semigroup S is abelian if f i • f j = f j • f i for all generators f i and f j of S. The transcendental semigroup S is bounded type (or finite type) if each of its generators f i is bounded type (or finite type).
The family F of complex analytic maps forms a normal family in a domain D if given any composition sequence (f α ) generated by the member of F , there is a subsequence (f α k ) which is uniformly convergent or divergent on all compact subsets of D. If there is a neighborhood U of the point z ∈ C such that F is normal family in U, then we say F is normal at z. If F is a family of members from the semigroup S, then we simply say that S is normal in the neighborhood of z or S is normal at z.
Let f be a transcendental entire map. We say that f iteratively divergent at Otherwise, a function f and semigroup S are said to be iteratively bounded at z. Note that in a semigroup In this sense, any f ∈ S is iteratively divergent at any z ∈ C always means that there is a sequence (g n ) n∈N in S representing g 1 = f, g 2 = f 2 , . . . , g n = f n , . . . such that g n (z) → ∞ as n → ∞ or there a sequence in S which contains (g n ) n∈N as a subsequence such that g n (z) → ∞ as n → ∞. More generally, semigroup S is iteratively divergent at any z ∈ C always means that every f ∈ S is iteratively divergent at z. That is, every sequence in S has a subsequence which diverges to infinity at z.
Based the Fatou-Julia-Eremenko theory, the Fatou set, Julia set and escaping set in the settings of transcendental semigroup are defined as follows. In our study, classical transcendental dynamics refers the iteration theory of single transcendental map and transcendental semigroup dynamics refers the dynamical theory generated by a set of transcendental entire maps. In transcendental semigroup dynamics, algebraic structure of semigroup naturally attached to the dynamics and hence the situation is largely complicated. The principal aim of this paper is to see how far classical transcendental dynamics applies to transcendental semigroup dynamics and what new phenomena appear in transcendental semigroup settings.
The fundamental contrast between classical transcendental dynamics and semigroup dynamics appears by different algebraic structure of corresponding semigroups. In fact, non-trivial semigroup need not be, and most often will not be abelian. However, trivial semigroup is cyclic and therefore abelian. As we discussed before, classical complex dynamics is a dynamical study of trivial (cyclic) semigroup whereas semigroup dynamics is a dynamical study of non-trivial transcendental semigroup.
The main motivation of this paper comes from seminal work of Hinkkanen and Martin [6] on the dynamics of rational semigroup and the extension study of K. K. Poon [11] to the dynamics of transcendental semigroup. 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 introduce fast escaping set in transcendental semigroup settings which is a main concern of our study. Note that the fast escaping set A(f ) in classical transcendental dynamics introduced first time by Bergweiler and Hinkkanen [3] and studied in more depth by Rippon and Stallard [14]. Indeed, it is a set consisting of points whose iterates tends to infinity as fast as possible and now plays a key role in classical transcendental dynamics. The set A(f ) has some properties exactly similar to those of I(f ).

Fast Escaping Set of Transcendental Semigroup
There is no equivalent formulation of fast escaping set in semigroup settings. We have started to define fast escaping set and try to formulate some other related terms and results. Note that it is our seminal work on the study of fast escaping set in transcendental semigroup settings.
Let S be a transcendental semigroup. Let us define a set of the form where M(r) = max |z|=r |f (z)|, with r R > 0 and M n (r) denotes the nth iterates of M(r) with itself. R > 0 can be taken any value such that M(r) > r for r R.
If r is sufficiently large then M n (r) → ∞ as n → ∞. The set A R (S) is well defined in semigroup S because for any f ∈ S, f n ∈ S for all n ∈ N. From the condition |f n (z)| M n (R) for all f ∈ S and n ∈ N of the set A R (S), we can also say that a point z ∈ C is in A R (S) if every sequence (g n ) n∈N in S has a subsequence (g n k ) n k ∈N which increases without bound at least as fast as the n-iterates of the maximum modulus of each g n k : M(r) = max |z|=r |g n k (z)|. Where g n k = f n such that |f n (z)| M n (R).
Definition 2.1 (Fast escaping set). The fast escaping set A(S) of a transcendental semigroup S consists the set A R (S) and all its pre-images. In other words, fast escaping set is the set of the form We can do certain stratification of fast escaping set which helps to make it more visible and provides a significant new understanding of the structure and properties of this set. We can write fast escaping set as a countable union of all its labels as we define below. Note that the set A R (S) defined above in 2.1 is the 0th level of fast escaping set A(S). Since M n+1 (R) > M n (R) for all n 0, so from 2.3, we get following chain of relation  We have mentioned several results and examples of transcendental semigroups in [16,17,18,20,21,22] where escaping set is non-empty. One of important particular result in this regards is a condition for which escaping set of a transcendental semigroup is same as escaping set of its each element. In such case, the fact would be obvious from classical transcendental dynamics that the fast escaping set is also non-empty. In our fourth coming study, we always talk with such a semigroup whose fast escaping set is non-empty.

Elementary Properties of Fast Escaping Set
In this section, we check how far basic properties of fast escaping set of classical transcendental dynamics can be generalize to fast escaping set of transcendental semigroup dynamics. In [17], we examined the contrast between classical complex dynamics and semigroup dynamics in the invariant features of Fatou set, Julia set and escaping set. In this paper, we see the same type contrast in fast escaping set. Note that in classical transcendental dynamics, the fast escaping set is completely invariant. We prove the following elementary results that are important regarding the structure of fast escaping set A(S). These results may also have more chances of leading further results concerning the properties and structure of A(S). Indeed, it shows certain connection and contrast between classical transcendental dynamics and transcendental semigroup dynamics and it is also a nice generalization of classical transcendental dynamics to semigroup dynamics.
Theorem 3.1. Let S is a transcendental semigroup such that A(S) = ∅. Then the following are hold.
(2) A(S) is independent of R.  (2) Choose R 0 > R, then from 2.4, we have A L R 0 (S) ⊂ A L R (S) for all L ∈ Z and so

Proof. (1). From the definition 2.2 (that is, from equation 2.3), we can write
Since there is m ∈ N such that M m (R) > R 0 and so From above two inequality, we have This proves A(S) is independent of R.  Proof. For each f ∈ S, Eremenko and Lyubich [5] proved that I(f ) ⊂ J(f ). K. K. Poon [11,Theorem 4.2] proved that J(S) = f ∈S J(f ). So for any f ∈ S, we have A(S) ⊂ A(f ) ⊂ J(f ) ⊂ J(S). The second part follows from A(S) ⊂ J(S) together with above theorem 3.1 (4).
There are many functions in the class B, the escaping set I(f ) consists of uncountable family of curves tending to infinity. For example, function λ sin z + γ with λ, γ ∈ C belongs to the class S ⊂ B and its escaping set is an uncountable union of curves tending to infinity, the so-called Cantor bouquet. For the function f (z) = λe z , 0 < λ < 1/e, the Fatou set is completely invariant attracting basin and Julia set is a Cantor bouquet consisting of uncountably many disjoint simple curves, each of which has finite end point and other endpoint is ∞.

On the L-th Labels of A(S)
In this section, we more concentrate on L-th label A L R (S) of fast escaping set A(S). Since fast escaping set can be written as expanding union of L-th labels, so we hope that structure and properties of each L-th label may determine structure and properties of fast escaping set. Again, we will also see contrast between fast escaping set and and its label if there are. The following result is a contrast. That is, analogous to classical transcendental dynamics [14], unlike the set A(S), each of its label is a closed set.     By part (2) of above theorem 2.1, we can conclude that U ⊂ A L R (S) for all simply connected component U of F (S). So, if all components of F (S) are simply connected, then we must ∂A L R (S) ⊂ J(S) and hence interior of A L R (S) is contained in F (S). This theorem also generalizes the result of classical transcendental dynamics to transcendental semigroup dynamics. That is, whatever Fatou component (simply or multiply connected) U of F (S) intersecting A(S), there is always U ⊂ A(S). Again, another question may raise. Such a Fatou component U is periodic or wandering? Note that in classical transcendental dynamics, such a Fatou component is always wandering ([14,Corollary 4.2]). For transcendental semigroup dynamics, such a Fatou component is again wandering domain. For, if U ∩ A(S) = ∅, then U ∩ A(f ) = ∅ for all f ∈ S. In this case U is wandering domain of each f ∈ S, so it is wandering domain of S as well.
Whatever discussion we have done above was about a Fatou component intersecting the fast escaping set A(S) of a transcendental semigroup S. Are there any Fatou components that are obviously known to lie in A(S)? In classical transcendental dynamics, its answer is yes (see for instance [14,Theorem 4.4] and [13,Theorem 2]). Indeed, in such case, the Fatou component that obviously lie in A(S) is a (closure of) multiply connected wandering domain. Bergweiler constructed an example of transcendental entire function f for which A(f ) contains simply connected wandering domain([2, Theorem 2]). This wandering domain is simply connected bounded one which lie in between multiply connected wandering domains and this one is only known example of non-multiply connected Fatou component that lie in A(f ). The generalization of above discussion to semigroup dynamics of course possible. For example, if U is a multiply connected wandering domain of F (S), then it also multiply connected wandering domains of every f ∈ S. In this case, U ⊂ A(f ) for all f ∈ S ( [14,Theorem 4.4]). Hence U ⊂ A(S).