aa r X i v : . [ m a t h . D S ] M a r ESCAPING SET OF HYPERBOLIC SEMIGROUP
BISHNU HARI SUBEDI AND AJAYA SINGH , Central Department of Mathematics, Institute of Science and Technology, TribhuvanUniversity, Kirtipur, Kathmandu, NepalEmail: [email protected], [email protected]
Abstract:
In this paper, we mainly study hyperbolic semigroups from which we get non-empty escaping setand Eremenko’s conjecture remains valid. We prove that if each generator of bounded type transcendentalsemigroup S is hyperbolic, then the semigroup is itself hyperbolic and all components of I ( S ) are unbounded . Key Words : Escaping set, Eremenko’s conjecture, transcendental semigroup, hyperbolic semigroup.
AMS (MOS) [2010] Subject Classification.
Introduction
Throughout this paper, we denote the complex plane by C and set of integers greaterthan zero by N . We assume the function f : C → C is transcendental entire function (TEF)unless otherwise stated. For any n ∈ N , f n always denotes the nth iterates of f . Let f bea TEF. The set of the form I ( f ) = { z ∈ C : f n ( z ) → ∞ as n → ∞} is called an escaping set and any point z ∈ I ( S ) is called escaping point . For TEF f , theescaping set I ( f ) was first studied by A. Eremenko [2]. He himself showed that I ( f ) = ∅ ;the boundary of this set is a Julia set J ( f ) (that is, J ( f ) = ∂I ( f )); I ( f ) ∩ J ( f ) = ∅ ; and I ( f )has no bounded component. By motivating from this last statement, he posed a question: Is every component of I ( f ) unbounded? . This question is considered as an important openproblem of transcendental dynamics and it is known as Eremenko’s conjecture . Note thatthe complement of Julia set J ( f ) in complex plane C is a Fatou set F ( f ).Recall that the set CV ( f ) = { w ∈ C : w = f ( z ) such that f ′ ( z ) = 0 } representsthe set of critical values . The set AV ( f ) consisting of all w ∈ C such that there exists acurve (asymptotic path) Γ : [0 , ∞ ) → C so that Γ( t ) → ∞ and f (Γ( t )) → w as t → ∞ iscalled 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 finitetype . 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 } This research work of first author is supported by PhD faculty fellowship from UniversityGrants Commission, Nepal.
BISHNU HARI SUBEDI, AJAYA SINGH are respectively called
Speiser class and
Eremenko-Lyubich class . The post-singular point is the point on the orbit of singular value. That is, if z is a singular value of entire function f , then f n ( z ) is a post-singular point for n ≥
0. The set of all post-singular points is called post-singular set and it is denoted by P ( f ) = [ n ≥ f n ( SV ( f ))The entire function f is called post-singularly bounded if its post-singular set is boundedand it is called post-singularly finite if its post-singular set is finite. A transcendental entirefunction f is hyperbolic if the post singular set P ( f ) is compact subset of the Fatou set F ( f ).The main concern of this paper is to study of escaping set under transcendental semi-group. So we start our formal study from the notion of transcendental semigroup. Notethat for given complex plane C , the set Hol( C ) denotes a set of all holomorphic functions of C . If f ∈ Hol( C ), then f is a polynomial or transcendental entire function. The set Hol( C )forms a semigroup with semigroup operation being the functional composition. Definition 1.1 ( Transcendental semigroup ) . Let A = { f i : i ∈ N } ⊂ Hol ( C ) be a set oftranscendental entire functions f i : C → C . A transcendental semigroup S is a semigroupgenerated by the set A with semigroup operation being the functional composition. Wedenote this semigroup by S = h f , f , f , · · · , f n , · · · i . Here, each f ∈ S is the transcendental entire function and S is closed under functionalcomposition. Thus f ∈ S is constructed through the composition of finite number offunctions f i k , ( k = 1 , , , . . . , m ). That is, f = f i ◦ f i ◦ f i ◦ · · · ◦ f i m .A semigroup generated by finitely many functions f i , ( i = 1 , , , . . . , n ) is called finitelygenerated transcendental semigroup . We write S = h f , f , . . . , f n i . If S is generated by onlyone transcendental entire function f , then S is cyclic or trivial transcendental semigroup .We write S = h f i . In this case each g ∈ S can be written as g = f n , where f n is the nthiterates of f with itself. The transcendental semigroup S is abelian if f i ◦ f j = f j ◦ f i for allgenerators f i and f j of S .Based on the Fatou-Julia-Eremenko theory of a complex analytic function, the Fatouset, Julia set and escaping set in the settings of semigroup are defined as follows. Definition 1.2 ( Fatou set, Julia set and escaping set ) . The set of normality or theFatou set of the transcendental semigroup S is defined by F ( S ) = { z ∈ C : S is normal in a neighborhood of z } The Julia set of S is defined by J ( S ) = C \ F ( S ) and the escaping set of S by I ( S ) = { z ∈ C : f n ( z ) → ∞ as n → ∞ for all f ∈ S } We call each point of the set I ( S ) by escaping point. It is obvious that F ( S ) is the largest open subset of C such that semigroup S is normal.Hence its compliment J ( S ) is a smallest closed set for any transcendental semigroup S . SCAPING SET OF HYPERBOLIC SEMIGROUP 3
Whereas the escaping set I ( S ) is neither an open nor a closed set (if it is non-empty) forany semigroup S .If S = h f i , then F ( S ) , J ( S ) and I ( S ) are respectively the Fatou set, Julia set and escap-ing set in classical iteration theory of complex dynamics. In this situation we simply write: F ( f ) , J ( f ) and I ( f ). For the existing results of Fatou Julia theory under transcendentalsemigroup, we refer [4, 5, 6, 8, 9].2. Some Fundamental Features of Escaping Set
The following immediate relation between I ( S ) and I ( f ) for any f ∈ S will be clearfrom the definition of escaping set. Theorem 2.1. I ( S ) ⊂ I ( f ) for all f ∈ S and hence I ( S ) ⊂ T f ∈ S I ( f ) .Proof. Let z ∈ I ( S ), then f n ( z ) → ∞ as n → ∞ for all f ∈ S . By which we mean z ∈ I ( f )for any f ∈ S . This immediately follows the second inclusion. (cid:3) Note that the above same type of relation (Theorem 2.1) holds between F ( S ) and F ( f ). However opposite relation holds between the sets J ( S ) and J ( f ). Poon [9, Theorem4.1, Theorem 4.2] proved that the Julia set J ( S ) is perfect and J ( S ) = S f ∈ S J ( f ) for anytranscendental semigroup S . From the last relation of above theorem 2.1, we can say thatthe escaping set may be empty. Note that I ( f ) = ∅ in classical iteration theory [2]. DineshKumar and Sanjay Kumar [5, Theorem 2.5] have mentioned the following transcendentalsemigroup S , where I ( S ) is an empty set. Theorem 2.2.
The transcendental entire semigroup S = h f , f i generated by two func-tions f and f from respectively two parameter families { e − z + γ + c where γ, c ∈ C and Re ( γ ) < , Re ( c ) ≥ } and { e z + µ + d, where µ, d ∈ C and Re ( µ ) < , Re ( d ) ≤ − } of functions hasempty escaping set I ( S ) . In the case of non-empty escaping set I ( S ), Eremenko’s result [2], ∂I ( f ) = J ( f ) ofclassical transcendental dynamics can be generalized to semigroup settings. The followingresults is due to Dinesh Kumar and Sanjay Kumar [5, Lemma 4.2 and Theorem 4.3] whichyield the generalized answer in semigroup settings. Theorem 2.3.
Let S be a transcendental entire semigroup such that I ( S ) = ∅ . Then (1) int ( I ( S )) ⊂ F ( S ) and ext ( I ( S )) ⊂ F ( S ) , where int and ext respectively denotethe interior and exterior of I ( S ) . (2) ∂I ( S ) = J ( S ) , where ∂I ( S ) denotes the boundary of I ( S ) . This last statement is equivalent to J ( S ) ⊂ I ( S ). If I ( S ) = ∅ , then we [11, Theorem4.6] proved the following result which is a generalization of Eremenko’s result I ( f ) ∩ J ( f ) = ∅ [2, Theorem 2] of classical transcendental dynamics to holomorphic semigroup dynamics. Theorem 2.4.
Let S be a transcendental semigroup such that F ( S ) has a multiply connectedcomponent. Then I ( S ) ∩ J ( S ) = ∅ BISHNU HARI SUBEDI, AJAYA SINGH
Eremenko and Lyubich [3] proved that if transcendental function f ∈ B , then I ( f ) ⊂ J ( f ), and J ( f ) = I ( f ). Dinesh Kumar and Sanjay Kumar [5, Theorem 4.5] generalizedthese results to a finitely generated transcendental semigroup of bounded type as shownbelow. Theorem 2.5.
For every finitely generated transcendental semigroup S = h f , f , . . . , f n i in which each generator f i is of bounded type, then I ( S ) ⊂ J ( S ) and J ( S ) = I ( S ) .Proof. Eremenko and Lyubich’s result [3] shows that I ( f ) ⊂ J ( f ) for each f ∈ S of boundedtype. Poon’s result shows [9, Theorem 4.2] that J ( S ) = S f ∈ S J ( f ). Therefore, (from thedefinition of escaping set and theorem 2.1) for every f ∈ S, I ( S ) ⊂ I ( f ) ⊂ J ( f ) ⊂ J ( S ).The next part follows from the facts J ( S ) ⊂ I ( S ) and I ( S ) ⊂ J ( S ). (cid:3) Escaping set of Hyperbolic Semigroup
The definitions of critical values, asymptotic values and singular values as well as postsingularities of transcendental entire functions can be generalized to arbitrary setting oftranscendental semigroups.
Definition 3.1 ( Critical point, critical value, asymptotic value and singular value ) . A point z ∈ C is called critical point of S if it is critical point of some g ∈ S . A point w ∈ C is called a critical value of S if it is a critical value of some g ∈ S . A point w ∈ C is calledan asymptotic value of S if it is an asymptotic value of some g ∈ S . A point w ∈ C is calleda singular value of S if it is a singular value of some g ∈ S . For a semigroup S , if all g ∈ S belongs to S or B , we call S a semigroup of class S or B (or finite or bounded type). Definition 3.2 ( Post singularly bounded (or finite) semigroup ) . A transcendentalsemigroup S is said to be post-singularly bounded (or post-singularly finite) if each g ∈ S is post-singularly bounded (or post-singularly finite). Post singular set of post singularlybounded semi-group S is the set of the form P ( S ) = [ f ∈ S f n ( SV ( f )) Definition 3.3 ( Hyperbolic semigroup ) . An transcendental entire function f is said tobe hyperbolic if the post-singular set P ( f ) is a compact subset of F ( f ) . A transcendentalsemigroup S is said to be hyperbolic if each g ∈ S is hyperbolic (that is, if P ( S ) is a compactsubset of F ( S ) ). Note that if transcendental semigroup S is hyperbolic, then each f ∈ S is hyperbolic.However, the converse may not true. The fact P ( f k ) = P ( f ) for all k ∈ N shows that f k ishyperbolic if f is hyperbolic. The following result has been shown by Dinesh Kumar andSanjay Kumar [5, Theorem 3.16] where Eremenko’s conjecture holds. Theorem 3.1.
Let f ∈ B periodic with period p and hyperbolic. Let g = f n + p, n ∈ N .Then S = h f, g i is hyperbolic and all components of I ( S ) are unbounded. SCAPING SET OF HYPERBOLIC SEMIGROUP 5
Example 3.1. f ( z ) = e λz is hyperbolic entire function for each λ ∈ (0 , e ) . The semigroup S = h f, g i where g = f m + p , where p = πiλ , is hyperbolic transcendental semigroup. We have generalized the above theorem 3.1 to finitely generated hyperbolic semigroupwith some modifications. This theorem will be the good source of non-empty escaping settranscendental semigroup from which the Eremenko’s conjecture holds.
Theorem 3.2.
Let S = h f , f , . . . , f n i is an abelian bounded type transcendental semigroupin which each f i is hyperbolic for i = 1 , , . . . , n . Then semigroup S is hyperbolic and allcomponents of I ( S ) are unbounded. Lemma 3.1.
Let f and g be transcendental entire functions. Then SV ( f ◦ g ) ⊂ SV ( f ) ∪ f ( SV ( g )) .Proof. See for instance [1, Lemma 2]. (cid:3)
Lemma 3.2.
Let f and g are permutable transcendental entire functions. Then f m ( SV ( g )) ⊂ SV ( g ) and g m ( SV ( f )) ⊂ SV ( f ) for all m ∈ N .Proof. We first prove that f ( SV ( g )) ⊂ SV ( g ). Then we use induction to prove f m ( SV ( g )) ⊂ SV ( g ).Let w ∈ f ( SV ( g )). Then w = f ( z ) for some z ∈ SV ( g ). In this case, z is either criticalvalue or an asymptotic value of function g .First suppose that z is a critical value of g . Then z = g ( u ) with g ′ ( u ) = 0. Since f and g are permutable functions, so w = f ( z ) = f ( g ( u )) = ( f ◦ g )( u ) = ( g ◦ f )( u ). Also,( f ◦ g ) ′ ( u ) = f ′ ( g ( u )) g ′ ( u ) = 0. This shows that u is a critical point of f ◦ g = g ◦ f and w is a critical value of f ◦ g = g ◦ f . By permutability of f and g , we can write f ′ ( g ( u )) g ′ ( u ) = g ′ ( f ( u )) f ′ ( u ) = 0 for any critical point u of f ◦ g . Since g ′ ( u ) = 0, theneither f ′ ( u ) = 0 ⇒ u is a critical point of f or g ′ ( f ( u )) = 0 ⇒ f ( u ) is a critical point of g .This shows that w = g ( f ( u )) is a critical value of g . Therefore, w ∈ SV ( g ).Next, suppose that z is an asymptotic value of function g . We have to prove that w = f ( z ) is also asymptotic value of g . Then there exists a curve γ : [0 , ∞ ) → C suchthat γ ( t ) → ∞ and g ( γ ( t )) → z . So, f ( g ( γ ( t ))) → f ( z ) = w as t → ∞ along γ . Since f ◦ g = g ◦ f , so f ( g ( γ ( t ))) → f ( z ) = w ⇒ g ( f ( γ ( t ))) → f ( z ) = w as t → ∞ along γ . Thisshows w is an asymptotic value of g . This proves our assertion.Assume that f k ( SV ( g )) ⊂ SV ( g ) for some k ∈ N with k ≤ m . Then f k +1 ( SV ( g )) = f ( f k ( SV ( g ))) ⊂ f ( SV ( g )) ⊂ SV ( g )Therefore, by induction, for all m ∈ N , we must have f m ( SV ( g )) ⊂ SV ( g ). The next part g m ( SV ( f )) ⊂ SV ( f ) can be proved similarly as above. (cid:3) Lemma 3.3.
Let f and g are two permutable hyperbolic transcendental entire functions.Then their composite f ◦ g is also hyperbolic. BISHNU HARI SUBEDI, AJAYA SINGH
Proof.
We have to prove that P ( f ◦ g ) is a compact subset of Fatou set F ( f ◦ g ). From [7,Lemma 3.2], F ( f ◦ g ) ⊂ F ( f ) ∩ F ( g ). This shows that F ( f ◦ g ) is a subset of F ( f ) and F ( g ). So this lemma will be proved if we prove P ( f ◦ g ) is a compact subset of F ( f ) ∪ F ( g ).By the definition of post singular set of transcendental entire function, we can write P ( f ◦ g ) = [ m ≥ ( f ◦ g ) m ( SV ( f ◦ g ))= [ m ≥ f m ( g m ( SV ( f ◦ g ))) (by using permutabilty of f and g ) ⊂ [ m ≥ f m ( g m ( SV ( f ) ∪ f ( SV ( g ))) (by above lemma 3.1)= [ m ≥ f m ( g m ( SV ( f ))) ∪ g m ( f m +1 ( SV ( g ))) ⊂ [ m ≥ f m ( SV ( f ))) ∪ [ m ≥ g m ( SV ( g ))) (by above lemma 3.2)= P ( f ) ∪ P ( g )Since f and g are hyperbolic, so P ( f ) and P ( g ) are compact subset of F ( f ) and F ( g ).Therefore, the set P ( f ) ∪ P ( g ) must be compact subset of F ( f ) ∪ F ( g ). (cid:3) Proof of the Theorem 3.2.
Any f ∈ S can be written as f = f i ◦ f i ◦ f i ◦ · · · ◦ f i m . Bypermutability of each f i , we can rearrange f i j and ultimately represented by f = f t ◦ f t ◦ . . . ◦ f t n n where each t k ≥ k = 1 , , . . . , n . The lemma 3.3 can be applied repeatablyto show each of f t , f t , . . . , f t n n is hyperbolic. Again by repeated application of above samelemma, we can say that f = f t ◦ f t ◦ . . . ◦ f t n n is itself hyperbolic and so the semigroup S ishyperbolic. Next part follows from [12, Theorem 3.3] by the assumption of this theorem. (cid:3) Acknowledgment : We express our heart full thanks to Prof. Shunshuke Morosawa,Kochi University, Japan for his thorough reading of this paper with valuable suggestionsand comments.
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