On the Hausdorff dimension of the escaping set of certain meromorphic functions

0.26 MB·English

by  Walter Bergweiler

ON THE HAUSDORFF DIMENSION OF THE ESCAPING SET OF
CERTAIN MEROMORPHIC FUNCTIONS
WALTER BERGWEILER AND JANINA KOTUS
9
0 Abstract. Letf be atranscendentalmeromorphicfunctionoffinite orderρ forwhich
0 the set of finite singularities of f−1 is bounded. Suppose that ∞ is not an asymp-
2 totic value and that there exists M ∈ N such that the multiplicity of all poles, except
n possibly finitely many, is at most M. For R > 0 let IR(f) be the set of all z ∈ C
a for which liminfn→∞|fn(z)| ≥ R as n → ∞. Here fn denotes the n-th iterate of f.
J Let I(f) be the set of all z ∈ C such that |fn(z)| → ∞ as n → ∞; that is, I(f) =

0 R>0IR(f). Denote the Hausdorff dimension of a set A ⊂ C by HD(A). It is shown
2
that limR→∞HD(IR(f))≤2Mρ/(2+Mρ). In particular, HD(I(f))≤2Mρ/(2+Mρ).
T
These estimates are best possible: for given ρ and M we construct a function f such
]
S that HD(I(f))=2Mρ/(2+Mρ) and HD(IR(f))>2Mρ/(2+Mρ) for all R>0.
D If f is as above but of infinite order, then the area of IR(f) is zero. This result does
. not hold without a restriction on the multiplicity of the poles
h
t
a
m
[ 1. Introduction and main results

1
v The Fatou set F(f) of a (non-linear) function f meromorphic in the plane is defined
4 as the set of all points z ∈ C such that the iterates fk of f are defined and form a normal
1
family in some neighbourhood of z. Furthermore, J(f) = C\F(f) where C = C∪{∞}
0
3 is called the Julia set of f and
.
1 I(f) = {z ∈ C : fn(z) → ∞ as n →b ∞} b
0
9
is called the escaping set of f. In addition to these sets, we shall also consider for R > 0
0
: the set
v
i I (f) = {z ∈ C : liminf|fn(z)| ≥ R}.
X R
n→∞
r Note that
a
I(f) = I (f).
R
R>0
\
It was shown by Eremenko [6] for entire f and by Dom´ınguez [5] for transcendental mero-
morphic f that I(f) 6= ∅ and that J(f) = ∂I(f). For an introduction to the iteration
theory of transcendental meromorphic functions we refer to [3]. Results on the Hausdorff
dimension of Julia sets and related sets are surveyed in [12, 21].
The set of singularities of the inverse function f−1 of f coincides with the set of critical
values and asymptotic values of f. We denote the set of finite singularities of f−1 by
sing(f−1). TheEremenko-Lyubich classB consistsofallmeromorphicfunctionsforwhich
sing(f−1) is bounded. Eremenko and Lyubich [7, Theorem 1] proved that if f ∈ B is
Date: January 20, 2009.
1991 Mathematics Subject Classification. 37F10 (primary), 30D05, 30D15 (secondary).
The authors were supportedby the EU ResearchTraining Network CODY. The firstauthor was also
supportedbytheG.I.F.,theGerman–IsraeliFoundationforScientificResearchandDevelopment,Grant
G-809-234.6/2003 and the ESF Research Networking Programme HCAA. The second author was also
supported by Polish MNiSW Grant N N201 0222 33 and PW Grant 504G 1120 0011 000.
1
2 WALTER BERGWEILER AND JANINAKOTUS
entire, then I(f) ⊂ J(f). This result was extended to meromorphic functions in B by
Rippon and Stallard [18]. Actually the proof yields that I (f) ⊂ J(f) if f ∈ B and R is
R
sufficiently large.
For A ⊂ C we denote by HD(A) the Hausdorff dimension of A and by area(A) the
two-dimensional Lebesgue measure of A. McMullen [15] proved hat HD(J(lez)) = 2 for
l ∈ C \ {0} and that area(J(sin(αz + βz))) > 0 for α,β ∈ C, α 6= 0. His proof shows
that the conclusion holds with J(·) replaced by I(·). Note that the functions considered
by McMullen are in the class B so that the escaping set is contained in the Julia set.
The order ρ(f) of a meromorphic function f is defined by
logT(r,f)
ρ(f) = limsup
logr
r→∞
where T(r,f) denotes the Nevanlinna characteristic of f; see [8, 9, 16] for the notations
of Nevanlinna theory. If f is entire, then we may replace T(r,f) by logM(r,f) here,
where M(r,f) = max |f(z)|. Thus for entire f we have [9, p. 18]
z=r
loglogM(r,f)
ρ(f) = limsup .
logr
r→∞
It is easy to see that ρ(lez) = ρ(sin(αz +βz)) = 1 for l,α,β ∈ C, l,α 6= 0.
McMullen’s result that HD(J(lez)) = 2 was substantially generalized by Baran´ski [1]
and Schubert [19] who proved that if f ∈ B is entire and ρ(f) < ∞, then HD(J(f)) = 2.
In fact, they show that HD(I (f)) = 2 for all R > 0 under these hypotheses. Their
R
proofs, which make use of the logarithmic change of variable introduced by Eremenko
and Lyubich, show that the conclusion holds more generally for meromorphic functions in
B which have finite order and for which ∞ is an asymptotic value. In fact, such functions
have a logarithmic singularity over ∞ and their dynamics are in many ways similar to
those of entire functions; see, e.g., [2] or [4].
Thepurposeofthispaperistoshowthatthesituationisverydifferent formeromorphic
functions of class B for which ∞ is not an asymptotic value.
Theorem 1.1. Let f ∈ B be a transcendental meromorphic function with ρ = ρ(f) < ∞.
Suppose that ∞ is not an asymptotic value and that there exists M ∈ N such that the
multiplicity of all poles, except possibly finitely many, is at most M. Then
2Mρ
(1.1) HD(I(f)) ≤
2+Mρ
and
2Mρ
(1.2) lim HD(I (f)) ≤ .
R
R→∞ 2+Mρ
Note that I (f) ⊂ I (f) if S > R. Hence HD(I (f)) is a non-increasing function of
S R R
R and thus the limit in (1.2) exists. Clearly (1.1) follows from (1.2) so that it suffices to
prove (1.2).
We note that elliptic functions are in B and have order 2. It was shown in [11, Theo-
rem 1.2] that if M denotes the maximal multiplicity of the poles of an elliptic function f,
then HD(I(f)) ≤ 2M/(1+M). Inequality (1.1) generalizes this result.
On the other hand, it was shown in [10, Example 3] that if f is an elliptic function such
that the closure of the postcritical set is disjoint from the set of poles, then HD(J(f)) ≥
2M/(1 + M). The argument shows that HD(I(f)) ≥ 2M/(1 + M). Thus (1.1) is best
ON THE HAUSDORFF DIMENSION OF THE ESCAPING SET 3
possible if ρ = 2. The following result shows that Theorem 1.1 is best possible for all
values of ρ.
Theorem 1.2. Let 0 < ρ < ∞ and M ∈ N. Then there exists a meromorphic function
f ∈ B of order ρ for which all poles have multiplicity M and for which ∞ is not an
asymptotic value such that
2Mρ
(1.3) HD(I(f)) =
2+Mρ
and
2Mρ
(1.4) HD(I (f)) >
R
2+Mρ
for all R > 0.
For functions of infinite order we cannot expect the Hausdorff dimension of J(f) or
I (f) to be less than 2. However, we have the following result.
R
Theorem 1.3. Let f ∈ B be a transcendental meromorphic for which ∞ is not an asymp-
totic value. Suppose that there exists M ∈ N such that all poles of f have multiplicity at
most M. Then
area(I (f)) = 0
R
for sufficiently large R. In particular,
area(I(f)) = 0.
The proof of Theorem 1.3 uses well-known techniques, see [20] for a similar argument.
In fact, as kindly pointed out to us by Lasse Rempe, Theorem 1.3 is implicitly contained
in [17, Theorem 7.2]. However, we shall include the short proof of Theorem 1.3 for
completeness.
Finally we show that the hypothesis on the multiplicity of the poles is essential.
Theorem 1.4. There exists a transcendental meromorphic f ∈ B for which ∞ is not an
asymptotic value and for which
area(I(f)) > 0.
2. Notations and preliminary Lemmas
The diameter of a set K ⊂ C is denoted by diam(K). Later we will also use the area
and diameter with respect to the spherical metric χ. We will denote them by area (K)
χ
and diam (K), respectively.
χ
For a ∈ C and r,R > 0 we use the notation D(a,r) = {z ∈ C : |z −a| < r} and
B(R) = {z ∈ C : |z| > R}∪{∞}.
The following lemma is known as Koebe’s distortion theorem and Koebe’s 1-theorem.
4
4 WALTER BERGWEILER AND JANINAKOTUS
Lemma 2.1. Let g : D(a,r) → C be univalent, 0 < λ < 1 and z ∈ D(a,λr). Then
λ λ
(2.1) |g′(a)|r ≤ |g(z)−g(a)| ≤ |g′(a)|r,
(1+λ)2 (1−λ)2
1−λ 1+λ
(2.2) |g′(a)| ≤ |g′(z)| ≤ |g′(a)|r
(1+λ)3 (1−λ)3
and
(2.3) g(D(a,r)) ⊃ D g(a), 1|g′(a)|r .
4
(cid:0) (cid:1)
Koebe’s theorem is usually only stated for the special case that a = 0, r = 1, g(0) = 0
and g′(0) = 1, but the above version follows immediately from this special case.
The following result is due to Rippon and Stallard [18, Lemma 2.1].
Lemma 2.2. Let f ∈ B be transcendental. If R > 0 such that sing(f−1) ⊂ D(0,R), then
all components of f−1(B(R)) are simply-connected. Moreover, if ∞ is not an asymptotic
value of f, then all components of f−1(B(R)) are bounded and contain exactly one pole
of f.
The following result is known as Iversen’s theorem ([8, p. 171] or [16, p. 292]).
Lemma 2.3. Let f be a transcendental meromorphic function for which ∞ is not an
asymptotic value. Then f has infinitely many poles.
Let (a ) be a sequence of non-zero complex numbers such that lim |a | = ∞. Then
j j→∞ j
∞
σ = σ((a )) = inf t > 0 : |a |−t < ∞
j j
( )
j=1
X
is called the exponent of convergence of the sequence (a ). Here we use the convention
j
that inf∅ = ∞, meaning that σ = ∞ if ∞ |a |−t = ∞ for all t > 0.
j=1 j
P
The following lemma is standard [9, p. 26].
Lemma 2.4. Let f be a transcendental meromorphic function and let σ be the exponent
of convergence of the non-zero poles of f. Then σ ≤ ρ(f).
We mention that a result of Teichmu¨ller [22] says that if f ∈ B is transcendental, if ∞
is not an asymptotic value of f and if there exists M ∈ N such that all poles of f have
multiplicity at most M, then m(r,f) = O(1) as r → ∞. This easily implies that the
exponent of convergence of the non-zero poles of f is actually equal to ρ(f) in this case.
ON THE HAUSDORFF DIMENSION OF THE ESCAPING SET 5
3. Proof of Theorem 1.1
ByLemma 2.3, f hasinfinitely manypoles. Let(a )bethesequence ofpolesf,ordered
j
such that |a | ≤ |a | for all j, and let m be the multiplicity of a . Then
j j+1 j j
b mj
j
f(z) ∼ as z → a
j
z −a
(cid:18) j(cid:19)
for some b ∈ C\{0}. We may assume that |a | ≥ 1 for all j ∈ N. Let R > 1 such that
j j 0
sing(f−1) ⊂ D(0,R ) and |f(0)| < R .
0 0
Lemma 2.2 says that if R ≥ R , then all components of f−1(B(R)) are bounded
0
and simply-connected and each component contains exactly one pole. We denote the
component containing a by U and choose a conformal map φ : U → D(0,R−1/mj)
j j j j
satisfying φ (a ) = 0. Then |f(z)φ (z)mj| → 1 as z approaches the boundary of U .
j j j j
Since |f(z)φ (z)mj| remains bounded near a and is non-zero in U , we deduce from the
j j j
maximum principle that |f(z)φ (z)mj| = 1 for all z ∈ U \{a } and that |φ′(a )| = 1/|b |.
j j j j j j
We may actually normalize φ such that φ′(a ) = 1/b . Denote the inverse function of
j j j j
φ by ψ . Since ψ (0) = a and ψ′(0) = b we deduce from (2.3) that
j j j j j j
1 1
(3.1) U = ψ (D(0,R−1/mj)) ⊃ D a , |b |R−1/mj ⊃ D a , |b | .
j j j j j j
4 4R
(cid:18) (cid:19) (cid:18) (cid:19)
Since |f(0)| < R we have 0 ∈/ U . Thus (3.1) implies in particular that
j
1
|b | ≤ |a |
j j
4R
for all R ≥ R and hence that
0
(3.2) |b | ≤ 4R |a |.
j 0 j
We note that ψ actually extends to a map univalent in D(0,R−1/mj). Applying (2.1)
j 0
with
R −1/mj R 1/mj
0
λ = =
R R
(cid:18) 0(cid:19) (cid:18) (cid:19)
we find that
λ
U ⊂ D a , |b |R−1/mj .
j j (1−λ)2 j
(cid:18) (cid:19)
Choosing R ≥ 2MR we have λ ≤ 1 and hence
0 2
(3.3) U ⊂ D a ,2|b |R−1/M ,
j j j
provided j is so large that m ≤ M. Co(cid:0)mbining (3.1) a(cid:1)nd (3.3) we thus have
j
1
D a , |b | ⊂ U ⊂ D a ,2R−1/M|b |
j j j j j
4R
(cid:18) (cid:19)
(cid:0) (cid:1)
for large j. Combining (3.2) and (3.3) we see that
U ⊂ D a ,8R |a |R−1/M .
j j 0 j
Choosing R ≥ (16R )M we thus have (cid:0) (cid:1)
0
1 3
(3.4) U ⊂ D a , |a | ⊂ D 0, |a | .
j j j j
2 2
(cid:18) (cid:19) (cid:18) (cid:19)
6 WALTER BERGWEILER AND JANINAKOTUS
Next we note that the U are pairwise disjoint. Combining this with (3.1) and (3.4) we
j
see that if n(r) denotes the number of a contained in the closed disc D(0,r), then
j
n(r) n(r)
π 1
|b |2 = area D a , |b |
16R2 j  j 4R j 
j=1 j=1 (cid:18) (cid:19)
X [
 
n(r)
≤ area U
j
 
j=1
[
 3
≤ area D 0, r
2
(cid:18) (cid:18) (cid:19)(cid:19)
9π
= r2.
4
Hence
n(r)
(3.5) |b |2 ≤ 36R2r2.
j
j=1
X
We shall use (3.5) to prove the following result.
Lemma 3.1. If
2Mρ
t > ,
2+Mρ
then
∞ t
|b |
j
< ∞.
|a |1+1/M
j=1 (cid:18) j (cid:19)
X
Proof. We put
ρ t t
s = −1 +1+ .
2 2 2M
(cid:18) (cid:19)
Then
ρ Mρ ρ
s > −1 +1+ = 1.
2 2+Mρ 2+Mρ
(cid:18) (cid:19)
For l ≥ 0 we put
P(l) = j ∈ N : n 2l ≤ j < n 2l+1 = j ∈ N : 2l ≤ |a | < 2l+1
j
and
(cid:8) (cid:0) (cid:1) (cid:0) (cid:1)(cid:9) (cid:8) (cid:9)
t t t(1−s+1/M)
|b | |b | 1
j j
S = = .
l |a |1+1/M |a |s |a |
j∈P(l)(cid:18) j (cid:19) j∈P(l)(cid:18) j (cid:19) (cid:18) j (cid:19)
X X
We now apply H¨older’s inequality, with p = 2/t and q = 2/(2−t). Putting
1 2 2Mρ+2
α = t 1−s+ = t > ρ
M 2−t 2M
(cid:18) (cid:19)
we obtain
t/2 (2−t)/2
|b |2 1
j
S ≤ .
l
 |a |2s  |a |α
j j
j∈P(l) j∈P(l)
X X
   
ON THE HAUSDORFF DIMENSION OF THE ESCAPING SET 7
Since α > ρ the series ∞ |a |−α converges by Lemma 2.4. Thus
j=1 j
P (2−t)/2
∞ (2−t)/2
1 1
≤ A := < ∞.
 |a |α |a |α
j j !
j∈P(l) j=1
X X
 
We now see, using (3.5), that
t/2
|b |2
j
S ≤ A
l  |a |2s
j
j∈P(l)
X
  t/2
1
≤ A |b |2
(2l)2s j 
j∈P(l)
X
A 
≤ 36R222(l+1) t/2
2lst
= A(12(cid:0)R)t 2t(1−s)(cid:1)l.
Since t(1−s) < 0, the series ∞ S conver(cid:0)ges. (cid:1) (cid:3)
l=0 l
Continuing with the proofPof Theorem 1.1 we note that in each simply-connected
domain D ⊂ B(R)\{∞} we can define all branches of the inverse function of f. Let g
j
be a branch of f−1 that maps D to U . Thus
j
1
(3.6) g (z) = ψ
j j z1/mj
(cid:18) (cid:19)
for some branch of the m -th root. We obtain
j
1 1
g′(z) = −ψ′ .
j j (cid:18)z1/mj(cid:19) mjz1+1/mj
Since we assumed that R ≥ 2MR we deduce from (2.2) with λ = 1 that
0 2
12|ψ′(0)| 12|b |
(3.7) |g′(z)| ≤ j = j ,
j |z|1+1/M |z|1+1/M
for z ∈ D ⊂ B(R) \ {∞}, provided j is so large that m ≤ M. From (3.3) we deduce
j
that
4
diam(U ) ≤ |b |.
k R1/M k
Moreover, if U ⊂ B(R), then
j
diamg (U ) ≤ sup |g′(z)|diamU
j k j k
z∈Uk
12|b | 4
j
≤ |b |
(1|a |)1+1/M R1/M k
2 k
4 |b |
= 21/M24 |b | k .
R1/M j |a |1+1/M
k
8 WALTER BERGWEILER AND JANINAKOTUS
Induction shows that if U ,U ,...,U ⊂ B(R), then
j1 j2 jl
diam g ◦g ◦...◦g (U )
j1 j2 jl−1 jl
(3.8) 4 |b | |b |
≤ (21/M2(cid:0)4(cid:0))l−1 |b | j2(cid:1) ..(cid:1). jl .
R1/M j1 |a |1+1/M |a |1+1/M
j2 jl
In order to obtain an estimate for the spherical diameter, we estimate the spherical
distance χ(z ,z ) of two points z ,z ∈ D(a , 1|a |). We have
1 2 1 2 j 2 j
2|z −z | 2|z −z | 8|z −z | 8|z −z |
1 2 1 2 1 2 1 2
χ(z ,z ) = ≤ ≤ ≤ .
1 2 1+|z |2 1+|z |2 1+ 1|a |2 1+|a |2 |a |1+1/M
1 2 4 j j j
Thus
p p
8
diam (K) ≤ diam(K)
χ
|a |1+1/M
j
for K ⊂ U and hence (3.8) yields
j
l
32 |b |
(3.9) diam g ◦g ◦...◦g (U ) ≤ (21/M24)l−1 jk .
χ j1 j2 jl−1 jl R1/M |a |1+1/M
k=1 jk
(cid:0)(cid:0) (cid:1) (cid:1) Y
Now there are m branches of the inverse function of f mapping U into U , for
jk jk+1 jk
k = 1,2,...,l−1. Overall we see that there are
l−1
m ≤ Ml−1
jk
k=1
Y
sets of diameter bounded as in (3.9) which cover all those components V of f−l(B(R))
for which fk(V) ⊂ U ⊂ B(R) for k = 0,1,...,l −1. We denote by E the collection
jk+1 l
of all components V of f−l(B(R)) for which fk(V) ⊂ B(R) for k = 0,1,...,l−1.
Next we note that (3.4) implies that if U ∩B(3R) 6= ∅, then |a | > 2R and U ⊂ B(R).
j j j
We conclude that E is a cover of the set
l
{z ∈ B(3R) : fk(z) ∈ B(3R) for 1 ≤ k ≤ l−1}.
Moreover, if t > 2Mρ/(2+Mρ), then
t ∞ ∞ l t
32 |b |
(diam (V))t ≤ Ml−1 (21/M24)l−1 ... jk
χ R1/M |a |1+1/M
VX∈El (cid:18) (cid:19) j1X=n(R) jl=Xn(R)kY=1(cid:18) jk (cid:19)
l
t ∞ t
1 32 |b |
= M(21/M24)t j .
M (2R)1/M24  |a |1+1/M 
(cid:18) (cid:19) j=n(R)(cid:18) j (cid:19)
X
Lemma 3.1 implies that  
∞ t
|b |
M(21/M24)t j < 1
|a |1+1/M
j=n(R)(cid:18) j (cid:19)
X
for large R. For such R we find that
lim (diam (V))t = 0
χ
l→∞
VX∈El
and thus
HD z ∈ B(3R) : fk(z) ∈ B(3R) for all k ∈ N ≤ t.
Hence HD(I (f)) ≤ t. As t > 2Mρ/(2+Mρ) was arbitrary, the conclusion follows.
3R (cid:0)(cid:8) (cid:9)(cid:1)
ON THE HAUSDORFF DIMENSION OF THE ESCAPING SET 9
4. Lower bounds for the Hausdorff dimension
In order to prove Theorem 1.2, we shall use results of Mayer [14] and McMullen [15].
ForsubsetsA,B oftheplane(orsphere) wedefinetheEuclideanandthesphericaldensity
of A in B by
area(A∩B) area (A∩B)
χ
dens(A,B) = and dens (A,B) = .
χ
area(B) area (B)
χ
Note that if
(4.1) B ⊂ {z ∈ C : R < |z| < S},
then
4 4
area(B) ≤ area (B) ≤ area(B)
(1+S2)2 χ (1+R2)2
and thus
1+R2 2 1+S2 2
(4.2) dens(A,B) ≤ dens (A,B) ≤ dens(A,B)
1+S2 χ 1+R2
(cid:18) (cid:19) (cid:18) (cid:19)
if B satisfies (4.1)
In order to state McMullen’s result, consider for l ∈ N a collection E of disjoint
l
compact subsets of C such that the following two conditions are satisfied:
(a) every element of E is contained in a unique element of E ;
l+1 l
(b) every elemenbt of E contains at least one element of E .
l l+1
Denote by E the union of all elements of E and put E = ∞ E . Suppose that (∆ )
l l l=1 l l
and (d ) are sequences of positive real numbers such that if B ∈ E , then
l l
T
dens (E ,B) ≥ ∆
χ l+1 l
and
diam (B) ≤ d .
χ l
Then we have the following result [15].
Lemma 4.1. Let E, E , ∆ and d be as above. Then
l l l
l+1 |log∆ |
j=1 j
limsup ≥ n−dimE.
|logd |
l→∞ P l
We remark that McMullen worked with the Euclidean density, but the above lemma
follows directly from his result.
We shall use Lemma 4.1 to prove (1.3). Of course, it follows from (1.3) that
2Mρ
(4.3) HD(I (f)) ≥ ,
R
2+Mρ
for all R > 0, but the application of Lemma 4.1 does not seem to yield (1.4), which says
that we have strict inequality in (4.3). However, in order to illustrate the method, we
shall first use Lemma 4.1 to prove (4.3). We will then describe the modifications that
have to be made in order to prove (1.3).
10 WALTER BERGWEILER AND JANINAKOTUS
The proof of (1.4) is based on the following result due to Mayer [14], which he obtained
using the theory of infinite iterated function systems developed by Mauldin and Ur-
ban´ski [13],
Lemma 4.2. Let f be a transcendental meromorphic function with ρ = ρ(f) < ∞.
Suppose that f has a pole a ∈ C\sing(f−1) and denote by M the multiplicity of a. Suppose
also that there are a neighbourhood D of a and constants K > 0 and α > −1−1/M such
that |f′(z)| ≤ K|z|α for z ∈ f−1(D). Then
ρ
(4.4) HD(J(f)) ≥ .
α+1+1/M
ActuallyMayer [14,Remark3.2]pointsoutthatif(z )denotesthesequence ofa-points
n
and if
∞
(4.5) |z |−ρ
n
n=1
X
diverges, then we have strict inequality in (4.4). Moreover, his proof shows that if f has
infinitely many poles a which satisfy the hypothesis of Lemma 4.2 and if the series (4.5)
diverges, then
ρ
(4.6) HD(I (f)) >
R
α+1+1/M
for each R > 0.
5. Construction of the example
In order to construct a function f to which the results of the previous section can be
applied we put µ = 2/ρ and define
∞ kµkzk
(5.1) g(z) = 2 .
z2k −k2µk
k=1
X
We note that if k ≥ (2|z|)1/µ, then
kµkzk kµk|z|k |z|k
≤ ≤ 2 ≤ 21−k.
z2k −k2µk k2µk −|z|2k kµk
(cid:12) (cid:12)
(cid:12) (cid:12)
Thus the series in (5.1) converges locally uniformly and hence it defines a function g
(cid:12) (cid:12)
meromorphic in C. Th(cid:12)e poles of g(cid:12)are at the points
u = kµexp(πil/k),
k,l
where k ∈ N and 0 ≤ l ≤ 2k −1. With v = kµ−1exp(πil(1−k)/k) we have
k,l
∞ 2k−1
v
k,l
g(z) = .
z −u
k,l
k=1 l=0
X X
Note that
(5.2) |v | = kµ−1 = |u |1−1/µ = |u |1−ρ/2.
k,l k,l k,l
We will show that g is bounded on the ’spider’s web’ W = W ∪W where
1 2
W = z : |z| = n+ 1 µ
1 2
n≥1
[ (cid:8) (cid:0) (cid:1) (cid:9)