ON LEVELS OF FAST ESCAPING SETS AND SPIDER’S WEB OF TRANSCENDENTAL ENTIRE FUNCTIONS

. Let f be a transcendental entire function and let I ( f ) be the points which escape to inﬁnity under iteration. Bergweiler and Hinkkanen introduced the fast escaping sets A ( f ) and subsequently, Rippon and Stallard introduced ‘Levels’ of fast escaping sets A LR ( f ). These sets under some restriction have the properties of “inﬁnite spider’s web” structure. Here we give some topological properties of the inﬁnite spider’s web and show some of the transcendental entire functions whose levels of the fast escaping sets have inﬁnite spider’s web structure.


Introduction
Let f : C −→ C be a transcendental entire function.For n ∈ N, f n denotes the n th iteration of f.Thus f n (z) = f (f n−1 )(z), where f 0 (z) = z and n = 1, 2, . . .A family of functions F is said to be a normal family, if every infinite sequence in the family has a subsequence which converges locally uniformly.The Fatou set F (f ) is defined to be the set of points z ∈ C, such that(f n ) n∈N forms a normal family in some neighbourhood of z.The complement of F (f ) denoted by J(f ) is called the Julia set.Clearly the Fatou set is open while the Julia set is closed.Also for a transcendental entire function it is known that the Julia set is unbounded.Baker [1] proved that the Julia set coincides with the closure of repulsive periodic points.For an introduction to the other properties of these sets one can refer, for instance, [2], [7].
For a transcendental entire function f, Eremenko [4] defined the escaping set as: and proved that I(f ) ∩ J(f ) = ∅, ∂I(f ) = J(f ) and all the components of I(f ) are unbounded.He further conjectured that all the components of I(f ) are unbounded.This led to a rich development of the field.Some partial results in the confirmation of this conjecture have been obtained, see for instance, [9], [10] and [11].
The rate at which the points of I(f ) escape to infinity also plays an important role.Bergweiler and Hinkkanen in [3] defined the fast escaping set, denoted by A(f ) as, Clearly A(f ) ⊂ I(f ).It was shown in [3] that A(f ) = ∅ and also that, ∂A(f ) = J(f ).
An alternative to A(f ), which has a geometric flavour was defined by Rippon and Stallard [10].They defined: 3) where D is any open disc meeting J(f ) and T (D) denotes the union of domain D with all of it's bounded complementary components.They showed that B(f ) is independent of D, completely invariant, B(f p ) = B(f ) for p ∈ N and finally B(f ) = A(f ).They further showed that all the components of A(f ) are unbounded and so I(f ) has at least one unbounded component.
In [11], Rippon and Stallard defined subsets of A(f ) called levels of A(f ), defined below.This lead to simplification of proofs of several earlier results and new insight into the properties of A(f ).See for instance [8], [11].
Definition 1.1.Let f be a transcendental entire function and R > 0 be such that M (r, f ) > r for r ≥ R. Let L ∈ Z, then the L th level (with respect to variable R) is defined to be Rippon and Stallard [11] proved the following: Here we supplement the above result by proving the next theorem, which deals with the levels of higher order.We observe that the above result cannot be generalized to each Level, i.e., the set relation f ) need not hold true for all n ∈ N. However we can prove the following: Theorem 1.1.Let f be a transcendental entire function.Let R > 0 be such that M (r, f ) > r for r ≥ R and L, n ∈ N then ).An interesting observation of the above theorem is that, if p is composite number having two different factorizations, say p = p 1 .q 1 = p 2 .q 2 , then Rippon and Stallard [11] first observed that A(f ), A R (f ) have interesting and intricate structure and called it as infinite spider's web, which is defined below.

Definition 1.2.
A set E is an (infinite) spider's web if E is connected and there exists a sequence of bounded simply connected domains Several examples of functions having A R (f ) as spider's web have been given.Sixsmith [13] gave examples of transcendental entire functions for which spider's web formation is there.Some work on spider's web has also been done by Helena and Peter [5] and Osborne [8].In their paper [11], Rippon and Stallard showed that for sufficiently large R, if A R (f ) c has a bounded component then each of A R (f ), A(f ), I(f ) is a spider's web.Also if f is a multiply connected Fatou component, then for sufficiently large R, A R (f ), A(f ), I(f ) are all spider's web.It is thus natural to look for relations between spider's web and levels of fast escaping sets, which we consider here.Also we study the topological properties of spider's web and show certain composite transcendental entire function h having the structure of A R (h) as spider's web.
We begin with a small but important observation that a bounded set can not be a spider's web.For suppose S is a bounded set and it also forms a spider's web, then there exits a positive constant K such that |z| ≤ K, for all z ∈ S, and there exits a sequence of bounded simply connected domain ) is a spider's web.The next Theorem deals with the union of spider's web.
Note that intersection of two spider's web need not form spider's web.Also continuous image of a spider's web need not form spider's web, for let f : C → C be a non constant continuous map defined by Then clearly f is a continuous map and if S be any spider's web, then f (S) is not a spider's web being bounded.However if we take f to be continuous open surjective map which maps bounded domain to bounded domain, then we have following theorem.In particular if f is a transcendental entire function without any exceptional value Picard, and without any asymptotic values, then also the image of a spider's web will be a spider's web.Corollary 1.1.Let f be a transcendental entire function with no finite asymptotic value and no exceptional value Picard.If S is a spider's web then so is f(S).

Proofs of Theorems on Levels of Fast Escaping Sets and Spider's Web
For the proof of Theorem 1.1 we need following lemmas.
Lemma 2.1.For any L, K ∈ N and sufficiently large R, The proof is an immediate consequence of Maximum modulus principle and the following lemma of Rippon and Stallard [11].
If R > 0 be sufficiently large then for n ∈ N From (2.3) and (2.4) we have The second set relation follows on similar lines.
For proving Theorem 1.2 we need following lemmas: As a consequence of Lemma 2.3 we have the following Lemma: Lemma 2.4.Let f be a transcendental entire function.Let R > 0 be such that We know by Theorem 1.
It is quite possible that there might exist more than one sequence of such domains say (H m ) m∈N satisfying the Definition 1.2.In order to distinguish the spider's web with corresponding domains we shall use the notation (E, G n ) n∈N and (E, H m ) m∈N respectively.
Proof of Theorem 1.3.Using induction it is sufficient to prove the theorem for two spider's webs.Let (S, G n ) n∈N and (T, H m ) m∈N , be two spider's webs.Now consider G 1 .Then there exits some Let T (D) denotes the union of domain D with all it's bounded complementary components, and define , and for n = 1, 2, . . ., G n and H k+(n−1) are simply connected and ) is simply connected.Thus K n is simply connected as well as bounded domain being union of two bounded domains.Further Thus (S ∪ T, K n ) n∈N is also a spider's web.
For proving the Theorem 1.4 we shall need the following lemma.
Proof.For (a), Let z ∈ T (A 1 ), if z ∈ A 1 then z ∈ A 2 ⊂ T (A 2 ).If z is in the bounded complementary component of A 1 , it is sufficient to show that z does not belong to any unbounded complementary component of A 2 .For suppose it does.Then there exits an arc in A c 2 joining z to ∞.As A 1 ⊂ A 2 , so this arc lies in A c 1 .Consequently z lies in unbounded complementary component of A 1 .This contradiction proves (a).The other results are simple set theoretic consequences of (a).

Note:
The conditions imposed on A 1 , A 2 in Lemma 2.5 are necessary, for instance, let Proof of Corollary 1.1.Being analytic, the function f is open and continuous.Also f would map bounded domain to a bounded domain, for suppose D is a bounded domain with f (D) unbounded, then there exists a curve Γ tending to ∞ in f (D) and consequently a curve γ in D tending to some α in D such that f (γ) = Γ, so that α is an asymptotic value for f , contradicting f by hypothesis, has no asymptotic value.
Next if z is not exceptional value Picard for f , then there exist (infinitely many) ξ ∈ C such that f (ξ) = z, and the proof now follows as in the previous theorem.

Bounded Fatou Components and Spider's Web
Regularity conditions and growth of transcendental entire function also play an important role in transcendental dynamics.In this section we discuss a result related to growth and regularity of transcendental entire function.We start with the following well known definitions of order ρ f and lower order λ f of entire functions respectively given by: ρ f = lim r→∞ log log M (r, f ) log r and λ f = lim r→∞ log log M (r, f ) log r .
Clearly we have 0 ≤ λ f ≤ ρ f ≤ ∞ and given any ρ, (0 ≤ ρ ≤ ∞), there exists an entire function of order ρ.We shall also need the following regularity condition: Further (see [13]) if a function f is log-regular then it satisfies Lemma 3.2 (b) (mentioned below) for all m > 1.
Lemma 3.1.[11] Let f be a transcendental entire function.Let R > 0 be such that M (r, f ) > r for r ≥ R and let A R (f ) be a spider's web, then f has no unbounded Fatou components Lemma 3.2.[13] Let f be a transcendental entire function.Let R > 0 be such that M (r, f ) > r for r ≥ R. Then A R (f ) is a spider's web if for some m > 1,

Theorem 1 . 4 .
If f : C −→ C be open continuous and a surjective map.Further let f map every bounded domain to bounded domain.If S be a spider's web, then f (S) is also a spider's web.

Proof of Theorem 1 . 4 .
Let (S, S n ) n∈N be a spider's web.Let H n = T (f (S n )) and denote H = f (S).Clearly H n are bounded simply connected domains being open continuous image of bounded simply connected domains.Now ∂H n by Lemma 2.5.Hence in order to show that f (S) is a spider's web it only remains that ∪ ∞ n=1 H n = C.For this consider any z ∈ C, then there exists some ζ ∈ C such that z = f (ζ), f being surjective.Also as ∪ n∈N S n = C, it follows that ζ ∈ S n for some n ∈ N and consequently

Definition 3 . 1 . [ 11 ]
Let c > 0. A transcendental entire function f is said to be log-regular with constant c, if the function φ(t) = M (e t , f ) satisfies φ