Non-existence of unbounded Fatou components of a meromorphic function
aa r X i v : . [ m a t h . C V ] N ov NON-EXISTENCE OF UNBOUNDED FATOUCOMPONENTS OF A MEROMORPHIC FUNCTION
ZHENG JIAN-HUA AND PIYAPONG NIAMSUP
Abstract
This paper is devoted to establish sufficient conditions under which atranscendental meromorphic function has no unbounded Fatou compo-nents and to extend some results for entire functions to meromorphicfunction. Actually, we shall mainly discuss non-existence of unboundedwandering domains of a meromorphic function. The case for a compo-sition of finitely many meromorphic function with at least one of thembeing transcendental can be also investigated in the argument of thispaper.
Keywords and Phases.
Fatou set, Julia set
Mathematics Subject Classification : 30D051.
Introduction and Main Results
Let M be the family of all functions meromorphic in the complexplane C possibly outside at most countable set, for example, a compo-sition of finitely many transcendental meromorphic functions is in M .Here we mean a function meromprphic in C with only one essentialsingular point at ∞ by a transcendental meromorphic function. Weshall study iterations of element in M . Mathematics Subject Classification.
Key words and phrases.
Transcendental meromorphic function, unbounded Fa-tou component, Julia set. Corresponding author.
We denote the n th iteration of f ( z ) ∈ M by f n ( z ) = f ( f n − ( z )) , n =1 , , . . . . Then f n ( z ) is well defined for all z ∈ C outside a (possible)countable set E ( f n ) = n − [ j =0 f − j ( E ( f )) , here E ( f ) is the set of all essential singular points of f ( z ). Define theFatou set F ( f ) of f ( z ) as F ( f ) = { z ∈ ¯ C : { f n ( z ) } is well definedand normal in a neighborhood of z } and J ( f ) = ¯ C \ F ( f ) is the Julia set of f ( z ). F ( f ) is open and J ( f )is closed, non-empty and perfect. It is well-known that both F ( f )and J ( f ) are completely invariant under f ( z ), that is, z ∈ F ( f ) ifand only if f ( z ) ∈ F ( f ) . And F ( f n ) = F ( f ) and J ( f n ) = J ( f ) forany positive integer n . We shall consider components of the Fatouset F ( f ) and hence let U be a connected component of F ( f ). Since F ( f ) is completely invariant under f , f n ( U ) is contained in F ( f ) andconnected, so there exists a Fatou component U n such that f n ( U ) ⊆ U n .If for some n ≥ , f n ( U ) ⊆ U , that is, U n = U , then U is calleda periodic component of F ( f ) and such the smallest integer n is theperiod of periodic component U . In particular, a periodic componentof period one is also called invariant. If for some n , U n is periodic, but U is not periodic, then U is called pre-periodic; A periodic component U of period p can be of the following five types: (i) attracting domainwhen U contains a point a such that f p ( a ) = a and | ( f p ) ′ ( a ) | < f np | U → a as n → ∞ ; (ii) parabolic domain when there exists apoint a ∈ ∂U such that f p ( a ) = a and ( f p ) ′ ( a ) = e πiα for α ∈ Q and f np | U → a as n → ∞ ; (iii) Baker domain when f np | U → a ∈ ∂U ∪ {∞} ON-EXISTENCE OF UNBOUNDED FATOU COMPONENTS 3 as n → ∞ and f p ( z ) is not defined at z = a ; (iv) Siegel disk when U is simply connected and contains a point a such that f p ( a ) = a and φ ◦ f p ◦ φ − ( z ) = e πiα z for some real irrational number α and aconformal mapping φ of U onto the unit disk with φ ( a ) = 0; (v)Hermanring when U is doubly connected and φ ◦ f p ◦ φ − ( z ) = e πiα z forsome real irrational number α and a conformal mapping φ of U onto { < | z | < r } . U is called wandering if it is neither periodic norpreperiodic, that is, U n ∩ U m = ∅ for all n = m. For the basic knowledgeof dynamics of a meromorphic function, the reader is referred to [5] andthe book [13].If for a function f ∈ M , f − ( E ( f )) contains at least three distinctpoints, then J ( f ) = ∞ [ n =1 f − n ( E ( f )) , and in any case, what we should mention is that for every n ≥ f n ( z )is analytic on F ( f ). In particular, this result holds for a compositionof finitely many meromorphic functions.Our study in this paper relies on the Nevanlinna theory of valuedistribution. To the end, let us recall some basic concepts and notationsin the theory. Let f ( z ) be a meromorphic function in C . Define m ( r, f ) = Z π log + | f ( re iθ ) | dθ and N ( r, f ) = Z r n ( t, f ) − n (0 , f ) t dt + n (0 , f ) log r, where n ( t, f ) is the number of poles of f ( z ) in the disk {| z | ≤ t } , and T ( r, f ) = m ( r, f ) + N ( r, f )which is known as the Nevanlinna characteristic function of f ( z ). Thequantity δ ( ∞ , f ) is the Nevanlinna deficiency of f at ∞ , defined by the ZHENG AND NIAMSUP following formula δ ( ∞ , f ) = lim inf r →∞ m ( r, f ) T ( r, f ) = 1 − lim sup r →∞ N ( r, f ) T ( r, f ) . (See [6]). The growth order and lower order of f ( z ) are defined respec-tively by λ ( f ) = lim sup r → + ∞ log T ( r, f )log r and µ ( f ) = lim inf r → + ∞ log T ( r, f )log r . In this paper, we take into account the question, raised by I. N. Bakerin 1984, of whether every component of F ( f ) of a transcendental entirefunction f ( z ) is bounded if its growth is sufficiently small. Baker [3]shown by an example that the order 1 / Theorem 1.1.
Let f ( z ) be a function in M . If we have (1) lim sup r → + ∞ L ( r, f ) r = + ∞ , where L ( r, f ) = min {| f ( z ) | : | z | = r } , then the Fatou set, F ( f ) , of f has no unbounded preperiodic or periodic components.In particular, f has no Baker domains. Theorem 1.1 confirms that an entire function whose growth doesnot exceed order 1 / ON-EXISTENCE OF UNBOUNDED FATOU COMPONENTS 5 than 1 / µ ( f ) < / δ ( ∞ , f ) > − cos( µ ( f ) π ),Theorem 1.1 also confirms that such a meromorphic function has nounbounded preperiodic or periodic components. And it is describedby an example in Zheng [15] that the condition (1) is sharpen. For acomposition g ( z ) = f m ◦ f m − ◦· · ·◦ f ( z ) of finitely many transcendentalmeromorphic functions f j ( z )( j = 1 , , . . . , m ; m ≥ F ( g ) has no unbounded periodic or preperiodiccomponents if for each j , there exits a sequence of positive real numberstending to infinity at which L ( r, f j ) > r and (1) holds for at least one f j .Therefore, the crucial point solving I. N. Baker’s question is in dis-cussion of non-existence of unbounded wandering domains of a mero-morphic function. There are a series of results for the case of entirefunctions on which some assumption on order less than 1 / f ( z ) be an transcen-dental entire function with order < /
2. Then every component of F ( f ) is bounded, provided that one of the following statements holds:(1) log M (2 r,f )log M ( r,f ) → c ≥ r → ∞ , (Stallard [11], 1993);(2) ϕ ′ ( x ) ϕ ( x ) ≥ cx , for all sufficiently large x , where ϕ ( x ) = log M ( e x , f )and c > M ( r m , f ) ≥ m log M ( r, f ) for each m > r (Hua and Yang [8], 1999);(4) µ ( f ) > c > c/ log 2 . However, an entire function with 0 < µ ≤ λ ( f ) < ∞ must ZHENG AND NIAMSUP satisfy the Hua and Yang’s assumption for m with µ ( f ) m > λ ( f ). Infact, choosing ε > µ − ε ) m > λ + 2 ε , we have for sufficientlylarge r > M ( r m , f ) > ( r m ) µ − ε > r ε r λ + ε ≥ r ε log M ( r, f ) . (2)What we should mention is that by modify a little the proof given in[8], Hua and Yang’s assumption for sufficiently large m instead of each m > Theorem 1.2.
Let f ( z ) be a transcendental entire function. If thereexists a d > such that for all sufficiently large r > we can find a ˜ r ∈ [ r, r d ] satisfying (3) log L (˜ r, f ) ≥ d log M ( r, f ) , then every component of F ( f ) is bounded. In [16] they also made a discussion of the case of composition of anumber of entire functions. In 2005, Hinkkanen [7] also gave a weakercondition than (3), that is, the coefficient ” d ” before log M ( r, f ) isreplaced by ” d (1 − (log r ) − δ )” with δ > Theorem 1.3.
Let f ( z ) be a transcendental meromorphic function andsuch that for some α ∈ (0 , and D > d > and all the sufficientlylarge r , there exists an t ∈ ( r, r d ) satisfying (4) log L ( t, f ) > αT ( r, f ) , j = 1 , , · · · , m. ON-EXISTENCE OF UNBOUNDED FATOU COMPONENTS 7 and (5) T ( r d , f ) ≥ DT ( r, f ) . Then F ( f ) has no unbounded components. Actually, the assumption in Theorem 1.3 is also a sufficient conditionof existence of buried points of the Julia set of a meromorphic functionwith at least one pole and which is not the form f ( z ) = a + ( z − a ) − p e g ( z ) . For such a meromorphic function, J ( f ) = S ∞ j =0 f − j ( ∞ ) andfrom Theorem 1.3 ∞ is a buried point of f ( z ) and therefore so are allprepoles.As a consequence of Theorem 1.3, we have the following Theorem 1.4.
Let f ( z ) be a transcendental meromorphic function with δ ( ∞ , f ) > − cos( πλ ( f )) and λ ( f ) < / and µ ( f ) > . Then F ( f ) has no unbounded compo-nents. In particular, Wang’s result can be deduced from Theorem .2.
The Proof of Theorems
To prove Theorems, we need some preliminary results. First prelim-inary result will be established by using the hyperbolic metric and ithas independent significance. To the end, let us recall some propertieson the hyperbolic metric, see ([1], [4]), etc. An open set W in C iscalled hyperbolic if C \ W contains at least two points (note ∞ hasbeen kicked out of W ). Let U be a hyperbolic domains in C . λ U ( z ) isthe density of the hyperbolic metric on U and ρ U ( z , z ) stands for the ZHENG AND NIAMSUP hyperbolic distance between z and z in U , i.e. ρ U ( z , z ) = inf γ ∈ U Z γ λ U ( z ) | dz | , where γ is a Jordan curve connecting z and z in U . For a hyperbolicopen set W , the hyperbolic density λ W ( z ) of W is the hyperbolic den-sity for each component of W . Then we convent that the hyperbolicdistance between two points which are in disjoint components equals to ∞ and the hyperbolic distance of two points a and b in one component U equals to ρ W ( a, b ) = ρ U ( a, b ). For a fixed point a W , introduce adomain constant C W ( a ) = inf {| z − a | λ W ( z ) : z ∈ W } . If U is simply-connected and d ( z, ∂U ) is a euclidean distance between z ∈ U and ∂U , then for any z ∈ U ,(6) 12 d ( z, ∂U ) ≤ λ U ( z ) ≤ d ( z, ∂U ) . Let f : U → V be analytic, where both U and V are hyperbolicdomains. By the principle of hyperbolic metric, we have(7) ρ V ( f ( z ) , f ( z )) ≤ ρ U ( z , z ) , for z , z ∈ U. In particular, if U ⊂ V , then λ V ( z ) ≤ λ U ( z ) for z ∈ U . Lemma 2.1. (cf. Zheng [13] ) Let U be a hyperbolic domain and f ( z ) a function such that each f n ( z ) is analytic in U and S ∞ n =0 f n ( U ) ⊂ W .If for some fixed point a W , C W ( a ) > and f n | U → ∞ , then forany compact subset K of U there exists a positive constant M = M ( K ) such that (8) M − | f n ( z ) | ≤ | f n ( w ) | ≤ M | f n ( z ) | for z, w ∈ K. ON-EXISTENCE OF UNBOUNDED FATOU COMPONENTS 9
Proof.
Under the assumption of Lemma 2.1, we obtain(9) ρ W ( z ) ≥ C W ( a ) | z − a | ≥ C W ( a ) | z | + | a | . It follows that ρ f n ( U ) ( f n ( z ) , f n ( w )) ≥ ρ W ( f n ( z ) , f n ( w )) ≥ C W ( a ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z | f n ( w ) || f n ( z ) | drr + | a | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = C W ( a ) (cid:12)(cid:12)(cid:12)(cid:12) log | f n ( z ) | + | a || f n ( w ) | + | a | (cid:12)(cid:12)(cid:12)(cid:12) . (10)Set A = max { λ U ( z, w ) : z, w ∈ K } . Clearly A ∈ (0 , + ∞ ). From(7), we have(11) ρ f n ( U ) ( f n ( z ) , f n ( w )) ≤ ρ U ( z, w ) ≤ A. Therefore, combining (10) and (11) gives(12) | f n ( z ) | + | a | ≤ ( | f n ( w ) | + | a | ) e A/C W ( a ) . This immediately completes the proof of Lemma 2.1.The following is Lemma of Zheng [15]( also see Theorem 1.6.7 of[13]).
Lemma 2.2.
Let f : U → U map the hyperbolic domain U ⊂ C analytically without fixed points and without isolated boundary pointsinto itself. If f n | U → ∞ ( n → ∞ ) , then for any compact subset K of U , we have (8) for some M = M ( K ) > . The following is a consequence of Lemma 2.1 and Lemma 2.2, whichis of independent significance.
Theorem 2.1.
Let f ( z ) be a function in M . If F ( f ) contains anunbounded component, then for any compact subset K of F ( f ) with f n | K → ∞ as n → ∞ , we have a positive constant M = M ( K ) suchthat (8) holds. Proof.
Assume without any loss of generalities that K is containedin a component U of F ( f ). If J ( f ) has one unbounded component, thenwe can find a subset Γ of J ( f ) such that C \ Γ is simply-connected.Then in view of Lemma 2.1 we shall get M = M ( K ) such that (8)holds by noting that S ∞ n =0 f n ( U ) ⊂ W = C \ Γ . Now assume that J ( f ) only has bounded components and thus F ( f )has only one unbounded component denoted by V . If S ∞ n =0 f n ( U ) doesnot intersect V , then in view of the fact that V has only boundedboundary components we can choose a path Γ in V tending to ∞ suchthat S ∞ n =0 f n ( U ) ⊂ W = C \ Γ . Thus as we did above, the result ofTheorem 2.1 follows.Let us consider the case when U ⊆ S ∞ n =0 f − n ( V ). If V is preperioidcor periodic, then an application of Lemma 2.2 yields the desired resultof Theorem 2.1; If V is wandering, then for some m > S ∞ n = m f n ( U )does not intersect V and therefore we can prove Theorem 2.1 in thiscase.The second preliminary result comes from the Poisson formula. Lemma 2.3.
Let f ( z ) be meromorphic on {| z | ≤ R } . Then thereexists a r ∈ ( R, R ) such that on | z | = r , we have (13) log + | f ( z ) | ≤ KT (3 R, f ) . where K ( ≤ is a universal constant, that is, it is independent of R, r and f . ON-EXISTENCE OF UNBOUNDED FATOU COMPONENTS 11
Proof.
Set D = {| z | ≤ R } . We denote by G D ( ζ , z ) the Greenfunction of D , that is, G D ( ζ , z ) = log (cid:12)(cid:12)(cid:12)(cid:12) (2 . R ) − ¯ zζ . R ( ζ − z ) (cid:12)(cid:12)(cid:12)(cid:12) , z, ζ ∈ D. A simple calculation implies that G D ( ζ , z ) ≤ log 5 R | ζ − z | and for ζ = 2 . Re iθ and r = | z | ≤ R , ∂∂~n G D ( ζ , z ) ds = Re 2 . Re iθ + z . Re iθ − z dθ ≤ . R + r . R − r dθ ≤ dθ. In view of the Poisson formula, we havelog | f ( z ) | = 12 π Z ∂D log | f ( ζ ) | ∂∂~n G D ( ζ , z ) ds − X a n ∈ D G D ( a n , z ) + X b n ∈ D G D ( b n , z ) ≤ m (2 . R, f ) + X b n ∈ D log 5 R | b n − z | , where a n is a zero and b n a pole of f ( z ) in D counted according to theirmultiplicities. According to the definition of N ( r, f ), we have n (2 . R, f ) ≤ (cid:18) log 65 (cid:19) − Z R . R n ( t, f ) t dt ≤ N (3 R, f ) . From the Boutroux-Cartan Theorem it follows that N Y n =1 | z − b n | ≥ (cid:18) R e (cid:19) N , N = n (2 . R, f ) , for all z ∈ C outside at most N disks ( γ ) the total sum of whosediameters does not exceed R/
2. Therefore there exists a r ∈ [ R, R ]such that {| z | = r } ∩ ( γ ) = ∅ and then on the circle | z | = r , we havelog + | f ( z ) | ≤ m (2 . R, f ) + N log 10 e < T (3 R, f ) . Thus we complete the proof of Lemma 2.3.
Proof Of Theorem 1.3.
For α >
0, there exists a natural number k such that D k − α ≥
1. Set h = d k . In view of (4) and (5), for all r ≥ R , we have a t ∈ ( r d k − , r h ) such thatlog L ( t, f ) ≥ αT ( r d k − , f ) ≥ αD k − T ( r, f ) ≥ T ( r, f ) , on | z | = t. (14)From Lemma 2.3, we have(15) log | f ( z ) | ≤ KT (3 r, f ) , for | z | ≤ r, where K is a positive constant independent of f and r .Take a positive integer m such that D ( m − k − > Kd mk = Kh m .Suppose that f has an unbounded Fatou component, say U . Assumethat U intersects | z | = R , otherwise we magnify R . Take a point z in U ∩{| z | = R } . Draw a curve γ ∈ U from z to U ∩{| z | = R H } , H = h m such that γ ⊂ {| z | = R H } except the end point of γ .Then there exists a z ∈ γ ∩ { R ≤ | z | ≤ R } such that log | f ( z ) | ≤ KT (3 R , f ). And there exists a r ∈ ( R h m − , R H ) such thatlog L ( r , f ) ≥ T ( R h m − , f ) = T ( R d ( m − k , f ) ≥ D ( m − k − T ( R d , f ) > Kh m T (3 R , f ) , (16)on | z | = r . Set R = exp( KT (3 R , f )). Then(17) f ( γ ) ∩ {| z | < R } 6 = ∅ and f ( γ ) ∩ {| z | > R H } 6 = ∅ . By the same argument as above, we have a z ∈ f ( γ ) ∩ { R ≤ | z | ≤ R } such that log | f ( z ) | ≤ KT (3 R , f ) and a r ∈ ( R h m − , R H ) suchthat log L ( r , f ) ≥ h m KT (3 R , f ) , on | z | = r . ON-EXISTENCE OF UNBOUNDED FATOU COMPONENTS 13
Set R = exp( KT (3 R , f )). Then since the circle {| z | = r } intersects f ( γ ), we have(18) f ( γ ) ∩ {| z | < R } 6 = ∅ and f ( γ ) ∩ {| z | > R H } 6 = ∅ . Define R n = exp( KT (3 R n − , f )) inductively. Then for each n > f n ( γ ) ∩ {| z | < R n } 6 = ∅ and f n ( γ ) ∩ {| z | ≥ R Hn } 6 = ∅ . Thus there is two points z n , w n ∈ γ such that(19) | f n ( z n ) | > R Hn > | f n ( w n ) | H . Combining (19) and Theorem 2.1 gives(20) | f n ( w n ) | H < | f n ( z n ) | ≤ M | f n ( w n ) | . This is impossible as n → ∞ , because a and e A are constants but H > | f n ( z n ) | → + ∞ as n → + ∞ . This completes the proof of Theorem 1.3.To prove Theorem 1.4, we need the following result, which was provedby Gol’dberg and Sokolovskaya [9].
Lemma 2.4.
Let f ( z ) be a transcendental meromorphic function with δ ( ∞ , f ) > − cos( πλ ( f )) and λ ( f ) < / . Then log dens E > , where E = { r > L ( r, f ) > αT ( r, f ) } for some positive α . In fact Lemma 2.4 asserts that for sufficiently large r >
0, we canfind a t ∈ [ r, r d ] for some d > L ( t, f ) > αT ( r, f ) . For a function f ( z ) with 0 < µ ( f ) ≤ λ ( f ) < + ∞ , we easily see thatlim r →∞ T ( r d , f ) T ( r, f ) = ∞ for d with dµ ( f ) > λ ( f ).Therefore Theorem 1.4 follows immediately from Theorem 1.3.3. Conclusion
By means of a careful calculation, indeed we can prove the followingresult: a transcendental meromorphic function f ( z ) has no unboundedcomponents of its Fatou set if for some 1 < d < D and all sufficientlylarge r there exists a t ∈ [ r, r d ] such thatlog L ( t, f ) > DT ( r, f ) . The argument of this paper is also available in establishing the corre-sponding results for a composition of finitely many meromorphic func-tions at least one of which is transcendental.
References [1] L. Alfors,
Conformal Invariants , McGram - Hill, New York, 1973.[2] J. M. Anderson and A. Hinkkanen, Unbounded domains of normality, Proc.Amer. Math. Soc., 126(198), 3243-3252.[3] I.N. Baker, The iteration of polynomials and transcendental entire functions,
J.Austral. Math. Soc. Ser.A , , 1981, 483–495.[4] A.F. Beardon and Ch. Pommerenke, The poincar` e metric of plane domains, J.London Soc., (2) , , 1978, 475 – 483. ON-EXISTENCE OF UNBOUNDED FATOU COMPONENTS 15 [5] W. Bergweiler, The iteration of meromorphic functions, Bull. Amer. Math. Soc.,(N.S.) 29(1993), 151-188.[6] W.K. Hayman,
Meromorphic functions , Oxford, 1964.[7] A. Hinkkanen, Entire functions with unbounded Fatou components, Contem-porary Math. 382(2005), 217-226.[8] X. Hua and C. Yang, Fatou components of entire functions of small growth,Erg. Th. Dynam. Sys., 19(1999), 1281-1293.[9] Gol’dberg and Sokolovskaya, Some relations for meromorphic functions of orderor lower order less than one, Izv. Vyssh. Uchebn. Zaved. Mat., 31 No. 6 (1987),26-31. Translation: Soviet Math. (Izv. VUZ) 31 No. 6(1987), 29-35.[10] G.M. Stallard, Some problems in the iteration of meromorphic functions, PhDThesis, Imperial College, London, 1991.[11] G.M. Stallard, The iteration of entire functions of small growth,
Math. Proc.Camb. Phil. Soc. , , 1993, 43–55.[12] Y. Wang, Bounded domains of the Fatou set of an entire function, Iarael J.Math. , , 2001, 55–60.[13] J. H. Zheng, Dynamics of Meromorphic Functions, Tsinghua University Press,2006. (in Chinese)[14] J.H. Zheng, Unbounded domians of normality of entire functions of smallgrowth, Math. Proc. Camb. Phil. Soc. , , 2000, 355–361.[15] J.H. Zheng ,On non-existence of unbounded domoins of normality of meromor-phic functions , J. Math. Anal. Appl. , (264) , 2001, 479–494.[16] J.H. Zheng and S. Wang , Boundedness of components of Fatou sets of en-tire and meromorphic functions, Indian J. Pure and Appl. Math. , 35 (10) ,2004,1137–1148.
Department of Mathematical Sciences, Tsinghua University, Bei-jing, 100084, P. R. China
E-mail address : [email protected] Department of Mathematics, Faculty of Science, Chiang Mai Uni-versity, Chiang Mai, 50200, THAILAND
E-mail address ::