The Dynamics of Semigroups of TranscendentalMeromorphic Functions

The Dynamics of Semigroups of Transcendental

Meromorphic Functions

HUANG 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, China

Abstract: 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 function

Introduction

ˆ(j=1,2,K)be transcendental mero-Letfj:C→C

morphic functions. We denote by

G = [f1, f2,K],

the semigroup generated by the family { fj: j=1,2,K} with the semigroup operation being functional compo-sition. We define the Fatou set of the semigroup G by

F(G) = {z∈C: G is defined and normal in some neighborhood of z}

ˆ\\F(G). The semi-and the Julia set of G byJ(G)=C

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) may

ˆwhen J(G) has an interior point, see the ex-not beC

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[1,2].

In the series of their papers, Hinkkanen and Martin

tried 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 Functions

Let G be a semigroup generated by a family of mero-ˆ, we define the backward morphic functions. Forz∈C

orbit O −(z) of z by

ˆ:there exists a g∈G such that g(w)=z} O −(z)={w∈C

and the exceptional set of G is defined by

ˆ: O−(z) is finite}. E(G)= {z∈C

Proposition 1 If z is not an element of E(G), then O−(z)⊇ J(G).

ˆ\\O−(z))⊆ Proof First of all, we prove g(C

ˆ\\O−(z)for any g∈G. Forx∈Cˆ\\O−(z),we need to C

ˆ\\O−(z).Suppose thatg(x)∉ prove that g(x)∈C

ˆ\\O−(z), then there exists a sequence xn∈O−(z) such C

Received: 2002-12-05; revised: 2003-11-11

E-mail: huang.z.g@263.sina.com; Tel: 86-512-66111539 ﹡﹡

that xn→g(x)(n→∞) and a sequence {gn} in G

HUANG Zhigang (黄志刚)：The Dynamics of Semigroups of Transcendental …… 473

satisfying g∞n{xn}=z. Thus, we havex∈{g−1(xn)}n=1. It implies thatx∈O−(z). It is a contradiction.

Sincez∉E(G),O−(z) have at least three points. Then C

ˆ\\O−(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 fm = amzn, m = 1,2,K, n ≥ 2 and |a|>1. Let

G = [f1, f2,K].

Then E(G)={0, ∞}. It is easy to see that 0 is a limit −mpoint of J(fm)={|z

|=|a|n−1}, and hence 0∈J(G).

In the classical case, for any component U of F( f ),

U1\\ f (U) contains at most one point, where U1 is a component of F( f ) with U1⊃f(U). However, this result cannot be preserved for a general semigroup.

Example 2 F(G) of the semigroup G = [zn, azn] with n > 2 and |a|>1 contains

Ω=⎧⎪nnn−1⎨1⎪⎩|a|<|z|<1⎫⎪|a|⎬ and

{|z|>1}. ⎪⎭By a simple calculation, forf=azn,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 z0 is called a repelling fixed point of G if there is an element f in G such that f(z0)=z0 and |f′(z0)|>1. And furthermore

J(G)=Uf∈GJ(f) (1) We can prove Proposition 2 by the similar argument

to those for the classical case of interaction of a single

function. Then Eq. (1) immediately follows from the

former result.

2 Semigroups of Transcendental

Entire Functions

We say a set M is forward invariant if f(M) ⊂ M for any f∈G and M is backward invariant if f−1(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 any

component 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 componentU

%of F(g) which contains U is multiply-connected. By Theorem

3.1 of Baker[3], the set {(U

%,gn):n=1,2,K} is infi-nite, and hence U is a wandering domain of G.

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 by

Sing(f−1)=Cri(f)UAsy(f).

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 a

family of transcendental meromorphic functions

f1,f2,...,fn.If each fj(1≤j≤n)∈B, then for all z∈F(G),

hm(z) does not tend to infinity, as m→∞, where hm(z)=fimofim−1o...ofi1(z),

and furthermore if , in addition, each fj is entire, then

gm(z)=fi1ofi2o...ofim(z)

does not tend to ∞ on F(G), 1≤ik≤n, 1≤k≤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 and0∉U∞f−s(∞). Then

s=1

there exist a positive constant R and a curve Γ connect-ing 0 and ∞ such that|f(z)|≤Ron Γ and for all

z∈C\\{0} which are not poles of f,

|f'(z)|≥|f(z)||f(z4|z|log)|

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 that0∉Un

U∞

fs=1s=1j−(∞).Then there are

jR and curves Γj(1≤j≤n) connecting 0 and ∞ such that|fj(z)|≤RonΓj(1≤j≤n)and for all z∈ C\\{0} which are not poles of fj, we have

|f|fj(z)|

j'(z)|≥

2π|z|log|fj(z)|R, 1≤j≤n. Suppose that there are a point z0∈F(G) and a set

{i1,i2,...}, 1≤ik≤n such that

hm(z0)=fimoKofi1(z0)→∞, as m→∞. Since {hm(z)} is normal at z0, we can take a fixed positive number R0 such thathm|B(z0,R0)→∞,and for m ≥m0≥0,hm(B(z0,R0))⊂F(G)I{|z|>R},where B(z0, R0) denotes the disk with center z0 and radius R0. We can assume m0=1 without any loss of generality. Lemma 1 implies

hm:B(z0,R0)→C\\Γq, q=im+1. By the principle of hyperbolic metric[7], we have

λC\\Γq[hm(z0)]|h′m(z1

0)|≤R, 0whereλC\\Γq(z)is the hyperbolic density on C\\Γq. Since

C\\Γq is simply connected, it is easy from the Koebe1

4

Theorem to prove that

inf{λz∈C\\Γ1

C\\Γ(z)dist(z,Γq):∀q}≥,

q4

where dist(z, Γq) is the Euclidean distance of z to Γ q. Therefore,

|h'(z44

m0)|≤Rdist(hm(z0),Γq)≤|hm(z0)| (3)

0R0

Put wp=hp(z0),p=1,2,K,w0=z0. Since wp∈F(G), wp is not a pole of any element of G. Then by Formula (2), we have

|f|fj(wp)||fj(wp)|

j'(wp)|≥2π|wlog, p|Rand therefore,

|h'm−1

m(z0)|=∏|f'i1|k+1(wm−1

|wk)|≥k+log|wk+1|

k=0k∏

=02π|wk|

R=

|hm(z0)|m∏−11log|wk+1|

|z. 0|k=02πR

This inequality contradicts Formula (3) since mk∏−11log|wk+1|

→∞(m→∞).

=02πRIf each fj∈B is entire, every component of F(G) is

simply connected.hm(B(z0,R0))is contained in a sim-ply connected component of F(G). We can deduce Formula (3) by the same method. Theorem 2 follows. References

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