The Dynamics of Semigroups of TranscendentalMeromorphic Functions

TSINGHUA SCIENCE AND TECHNOLOGY ISSN 1007-0214 18/21 pp472-474 Volume 9, Number 4, August 2004)be The Dynamics of Semigroups of TranscendentalMeromorphic FunctionsHUANG Zhigang (黄志刚)**Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China;Department of Applied Mathematics, University of Science and Technology of Suzhou, Suzhou 215009, ChinaAbstract: This paper considers the dynamics associated with an arbitrary semigroup of transcendental meromorphic functions. Fatou-Julia theory was used to investigate the dynamics of these semigroups. Some results of the dynamics of a rational mapping on the Riemann sphere were extended to the case. Key words: semigroup; dynamics; transcendental entire functionIntroductionLet transcendental mero-morphic functions. We denote byˆ:(1,2,j f C Cj →=K G = [f 1, f 2,K ],the semigroup generated by the family { f j : j =1,2,K } with the semigroup operation being functional compo-sition. We define the Fatou set of the semigroup G byF (G ) = {z ∈C : G is defined and normal in some neighborhood of z }and the Julia set of G by . The semi-group generated by a single function f is denoted by [ f ]. We write F ( f ) for F ([ f ]) and J ( f ) for J ([ f ]). Then F ( f ) and J ( f ) are the Fatou set and Julia set, re-spectively, in the Fatou-Julia classical iteration theory. It is obvious that the dynamics of a semigroup is more complicated than that of the iteration of a single func-tion. Some properties in the classical case cannot be preserved for the semigroup case. For example, F (G ) and J (G ) may not be complete invariant and J (G ) maynot be when J (G ) has an interior point, see the ex-amples in Ref. [1]. We can also list some same proper-ties of the dynamics of a semigroup as that for iteration of a single function ˆ()\()J G CF G =ˆC[1,2].In the series of their papers, Hinkkanen and Martintried to extend the classical theory of the dynamics as-sociated to the iteration of a rational function of a complex variable to the more general setting of an arbi-trary semigroup of rational functions [1,2]. In this paper, we will extend some results of the classical theory to the semigroup of meromorphic functions.1 Some Properties and Exceptional Sets of a Semigroup Generated by a Family of Meromorphic FunctionsLet G be a semigroup generated by a family of mero-morphic functions. For ˆz C∈, we define the backward orbit O −(z ) of z byO −(z )={there exists a g ∈G such that g (w )=z } ˆ:w C∈and the exceptional set of G is defined byE (G )= {ˆz C∈: O −(z ) is finite}. Proposition 1 If z is not an element of E (G ), then ()O z −⊇ J (G ).Proof First of all, we prove ˆ(\())g CO z −⊆ ˆ\(CO z −)for any g ∈G . For ˆ\(),x C O z −∈we need to prove that ˆ()\().g x CO z −∈Suppose that ()g x ∉ ˆ\(CO z −)∞, then there exists a sequence x n ∈O −(z ) such that and a sequence {g ()()n x g x n →→n } in GReceived: 2002-12-05; revised: 2003-11-11﹡﹡ E-mail: huang.z.g@; Tel: 86-512-66111539HUANG Zhigang (黄志刚)：The Dynamics of Semigroups of Transcendental (473)satisfying g n {x n }=z . Thus, we have 11{()}n n x g x −=∈. It implies that ()x O z −∈. It is a contradiction.Since ()z E G ∉,()O z − have at least three points. Then ˆ\()(CO z F G −⊆), i.e., ()O z −()J G ⊇. Proposition 1 follows.Let G be generated by a family H of rational func-tions with the degree at least 2. If card(H )<∞, then E (G ) ⊂ F (G ). If card(H ) = ∞, we cannot assert E (G )⊂ F (G ). Example 1 Set f m = a m z n , m = 1,2,K , n ≥ 2 and |a |>1. LetG = [f 1, f 2,K ].Then E (G )={0, ∞}. It is easy to see that 0 is a limit point of 1(){||||m n m J f za −−==}, and hence 0∈J (G ).In the classical case, for any component U of F ( f ),U 1\ f (U ) contains at most one point, where U 1 is a component of F ( f ) with U 1⊃f (U ). However, this result cannot be preserved for a general semigroup.Example 2 F (G ) of the semigroup G = [z n , az n ] with n > 2 and |a |>1 contains||z Ω=<< and{|z |>1}. By a simple calculation, for ,n f az =we have f (Ω )⊂ {|z |>1}, and {|z |>1}\ f (Ω ) is an unbounded domain. Proposition 2 Let G be a semigroup generated by a family of meromorphic functions. Then J (G ) is per-fect and the closure of the repelling fixed points of G where z 0 is called a repelling fixed point of G if there is an element f in G such that f (z 0)=z 0 and 0|()|1f z ′>. And furthermore()()f G J G J f ∈=U (1) We can prove Proposition 2 by the similar argumentto those for the classical case of interaction of a singlefunction. Then Eq. (1) immediately follows from theformer result.2 Semigroups of TranscendentalEntire FunctionsWe say a set M is forward invariant if f (M ) ⊂ M for any f ∈G and M is backward invariant if 1()f M −= {:()}z f z M M ∈⊂ for any f ∈G . It is easy to deduce that F (G ) is forward invariant and J (G ) is backward invariant. Since F (G ) is forward invariant, for anycomponent U of F (G ) and any g ∈G ,()()g U F G ⊂, and g (U ) is contained in a component, denoted by (U , g ), of F (G ).Definition 1 A component U of F (G ) is called a wandering domain of G provided that the set {(U , g ): ∀g ∈G } is infinite; otherwise, U is non-wandering.Theorem 1 Let G be a semigroup generated by a family of transcendental entire functions. Then a mul-tiply-connected component of F (G ) must be wandering and bounded, in other words, a non-wandering compo-nent of F (G ) must be simply-connected.Proof Let U be a multiply-connected component of F (G ). We can draw a simple closed curve γ in U which is not null-holomotopic. Then from Proposition 2, there is g ∈G such that J (g ) intersects the bounded interior surrounded by γ. Since γ ⊂ U ⊂ F (g ), γ is not null-homotopic with respect to F (g ), otherwise ∞∈F (g ). This is impossible, and the component Uof F (g ) which contains U is multiply-connected. By Theorem3.1 of Baker %[3], the set {( is infi-nite, and hence U is a wandering domain of G .,):1,2,}n Ug n =%K We denote by Cri( f ) the set of critical values of f and by Asy( f ) the set of asymptotic values of f . Define the set of singularities of the inverse function of f bySing(1f −)Cri()Asy()f f =U .We denote by B the family of meromorphic func-tions with the bounded Sing(f −1). The method of proof of Theorem 2 in Ref. [4] was used, and the following theorem, which is the main result in this paper, can easily be proved. The idea of his method comes essen-tially from Eremenko and Lyubich [5] and Bergweiler [6], but Zheng [4] finished the proof of his Theorem 2 by combining the hyperbolic metric instead. Theorem 2 Let G be a semigroup generated by afamily of transcendental meromorphic functions12,,...,.n f f f If each f j (1≤j ≤n )∈B , then for all z ∈F (G ),h m (z ) does not tend to infinity, as m ∞, where →11()...()m m m i i i h z f f f z −=o o o ,and furthermore if , in addition, each f j is entire, then12()...()m m i i i g z f f f z =o o odoes not tend to ∞ on F (G ), , . 1k i n ≤≤1k m ≤≤Theorem 2 is an extension of Theorem 2 of Zheng [4]. In the proof of Theorem 2, we need the following result.474 Tsinghua Science and Technology , August 2004, 9(4): 472–474()∞Lemma 1 Suppose f ∈B and . Thenthere exist a positive constant R and a curve Γ connect-ing 0 and ∞ such that 10s s f ∞−=∉U |()|f z R ≤on Γ and for allwhich are not poles of f ,\{0}z C ∈ |()||'()|4||f z f z z ≥log |()|f z R (2)This first part of Lemma 1 is obvious and Inequality(2) was proved in Zheng [8] using the hyperbolic metric and in Ref. [9] for “16π” in place of “4” using the logarithmic change of variable in a neighborhood of infinity.Proof of Theorem 2 We can assume without any loss of generality that Then there areR and curves connecting 0 and ∞ such that 110ns j j s f ∞−==∉U U ().∞(1)j j n Γ≤≤|()|j f z R ≤on and for all z ∈ C\{0} which are not poles of f (1)j j n Γ≤≤j , we have|()|'()|2||j j |f z f z z ≥πlog |()j |f z R , 1. j n ≤≤Suppose that there are a point z 0∈F (G ) and a set, such that12{,,...}i i 1k i n ≤≤100()()m m i i h z f f z =→∞oKo m →∞, as . Since {h m (z )} is normal at z 0, we can take a fixed positive number R 0 such that and for m ≥m 00(,)|,m B z R h →∞0≥0,where B (z 00((,))(){||},m h B z R F G z R ⊂>I 0, R 0) denotes the disk with center z 0 and radius R 0. We can assume m 0=1 without any loss of generality. Lemma 1 implies00:(,)\m q h B z R C Γ→, . 1m q i +=By the principle of hyperbolic metric [7], we have\0001[()]|()|,q C m m h z h z R Γλ′≤ where \()q C z Γλis the hyperbolic density on C \Γq . SinceC \Γq is simply connected, it is easy from the Koebe 14Theorem to prove that\1inf{()dist(,):\}4q q q C z z z C ΓλΓΓ∀∈≥,where dist(z , Γq ) is the Euclidean distance of z to Γ q . Therefore,000044|'()|dist((),)|()|m m q h z h z h z R R Γ≤≤0m (3)Put w p =h p (z 0),1,2,,p =K w 0=z 0. Since w p ∈F (G ), w p is not a pole of any element of G . Then by Formula (2), we have|()||'()|2||j p j p p f w f w w ≥πlog |()|j p f w R , and therefore,1111000|||'()||'()|2||k m m k m i k k k k w h z f w w +−−+===≥π∏∏log 1||k w R +=1000|()|1||2m m k h z z −=π∏log 1||k w R+. This inequality contradicts Formula (3) since 1012m k −=π∏log 1||k w R +()m →∞→∞.If each f j ∈B is entire, every component of F (G ) issimply connected.is contained in a sim-ply connected component of F (G ). We can deduce Formula (3) by the same method. 00((,))m h B z R Theorem 2 follows. References[1] Hinkkanen A, Martin G J. The dynamics of semigroups ofrational functions I. Proc . London Math . Soc ., 1996, 73(3): 358-384.[2] Hinkkanen A, Martin G J. Julia sets of rational semigroups.Math. Z , 1996, 222: 161-169.[3] Baker I N. Wandering domains in the iteration of entirefunctions. Proc. London Math. Soc., 1984, 49: 563-576. [4] Zheng J H. Singularities and wandering domain in itera-tions of meromorphic functions. Illinois Math. J., 2000, 44(3): 520-530.[5] Eremenko A E, Lyubich M Y. Dynamical properties ofsome classes of entire functions. Ann. Inst. Fourier , 1992, 42: 989-1020.[6] Bergweiler W. Iterations of meromorphic functions. Bull.Amer. Math. Soc. (N .S ), 1993, 29: 151-188.[7] Zheng J H. Uniformly perfect sets and distortion of holo-morphic functions. Nagoya. Math. J., 2001, 164: 17-33. [8] Zheng J H. On transcendental meromorphic functionswhich are geometrically finite. J . Austral. Math. Soc ., 2002, 72: 93-107.[9] Rippon P J, Stallard G M. Iteration of a class of hyperbolicmeromorphic functions. Proc. Amer. Math. Soc., 1999, 207: 3251-3258.。