On fundamental loops and the fast escaping set

0.28 MB·English

by  Dave Sixsmith

ON FUNDAMENTAL LOOPS AND THE FAST ESCAPING SET
D.J.SIXSMITH
3
1
0 Abstract. The fast escaping set, A(f), of a transcendental entire function
2 f has begun to play a key role in transcendental dynamics. In many cases
A(f) has the structure of a spider’s web, which contains a sequence of fun-
n
damental loops. We investigate the structure of these fundamental loops for
a
J functions with a multiply connected Fatou component, and show that there
existtranscendentalentirefunctionsforwhichsomefundamentalloopsarean-
2
alytic curves andapproximately circles,whileothers aregeometricallyhighly
1
distorted. We do this by introducing a real-valued function which measures

therateofescapeofpointsinA(f),andshowthatthisfunctionhasanumber
]
S ofinterestingproperties.
D
.
h
at 1. Introduction
m
Suppose that f : C C is a transcendental entire function. The Fatou set
[ F(f) is defined as the s→et of points z C such that (fn)n∈N is a normal family
1 in a neighbourhood of z. The Julia set∈J(f) is the complement in C of F(f). An
v introduction to the properties of these sets was given in [5].
6 For a general transcendental entire function the escaping set
7
6 I(f)= z :fn(z) as n
{ →∞ →∞}
2
was studied first in [11]. This paper concerns a subset of the escaping set, called
.
1
the fast escaping set A(f). This was introduced in [7], and can be defined [21] by
0
3 (1.1) A(f)= z :there exists ℓ N such that fn+ℓ(z) Mn(R,f), for n N .
1 { ∈ | |≥ ∈ }
: Here,themaximummodulusfunctionM(r,f)=max|z|=r f(z),forr 0,Mn(r,f)
v | | ≥
denotes repeated iteration of M(r,f) with respect to the variable r, and R > 0 is
i
X such that Mn(R,f) as n . For simplicity, we only write down this
→ ∞ → ∞
r restrictiononR in formalstatements ofresults– elsewherethis shouldbe assumed
a to be true. We write M(r) when it is clear which function is being referred to.
In [21] several results on A(f) were proved by considering the closed sets
A (f)= z : fn(z) Mn(R), n N ,
R
{ | |≥ ∈ }
where R > 0 is such that Mn(R) as n . Rippon and Stallard [21] used
→∞ → ∞
propertiesofA (f)andA(f)todevelopnewresultsrelatingtoEremenko’sconjec-
R
ture [11] that I(f) contains no bounded components. In addition they introduced
the conceptofa spider’s web. AsetE is aspider’s webifE is connectedandthere
exists a sequence of bounded simply connected domains (Gn)n∈N such that
∂G E, G G , for n N, and G = C.
n n n+1 n
⊂ ⊂ ∈
n∈N
[
A transcendental entire function for which A (f) is a spider’s web has very
R
strong dynamical properties – for example, it is shown in [21] that A(f) and I(f)
1
2 D.J.SIXSMITH
arealsospiders’webs,Eremenko’sconjectureholdsandallcomponentsoftheFatou
setarebounded. Therearemanylargeclassesoftranscendentalentirefunctionsfor
whichit is knownthat A (f) is a spider’sweb; see [21], [17] and[22] for examples.
R
To understand the structure of A (f) spiders’ webs, Rippon and Stallard [21]
R
introduced fundamental holes and fundamental loops. When A (f) is a spider’s
R
web,wedefinethefundamentalholeH asthecomponentofA (f)c thatcontains
R R
the origin, and the fundamental loop L by L =∂H . Since A (f) is closed, we
R R R R
have that L A (f).
R R
⊂
Ournotationherediffersslightlyfromthatin[21]. ForR>0fixed,Ripponand
Stallard define sets
Am(f)= z : fn(z) Mn+m(R), n N, n+m 0 , for m 0,
R { | |≥ ∈ ≥ } ≥
anddefinethe sequenceoffundamentalholesto bethe componentsofAm(f)c that
R
contain the origin. Denoting this sequence by (H′ ) , we observe that these
m m≥0
notations are related by the equation
H =H′ , for m 0.
Mm(R) m ≥
Itwasshownin[21,Theorem1.9(a)]thatA (f)isaspider’swebwheneverf is
R
a transcendental entire function with a multiply connected Fatou component. We
now give the first results on the properties of fundamental loops in this case. The
first of these gives information on the location of some fundamental loops. We say
thatasetU surroundsasetV ifandonlyifV iscontainedinaboundedcomponent
of C U. We also write dist(z,U)=inf z w.
w∈U
\ | − |
Theorem 1.1. Suppose that f is a transcendental entire function. Then there
exists R′ = R′(f) > 0 such that the following holds. If U is a multiply connected
Fatou component of f, such that U surrounds the origin and dist(0,U)>R′, then
there exist 0<R <R such that
1 2
(a) L =∂ U;
R1 int
(b) L =∂ U;
R2 out
(c) if L is a fundamental loop such that L U = , then L U. Moreover,
R R R
∩ 6 ∅ ⊂
this condition occurs if and only if R <R<R .
1 2
Here ∂ U is defined as the boundary of the unbounded component of C U,
out
and ∂ U is defined as the boundary of the component of C U that contains\the
int
\
origin. The related set ∂ U is defined in [8] as the boundary of the component of
inn
C U that contains the origin.
\
In general, if U is a Fatou component, we write U , n 0, for the Fatou
n
≥
component containing fn(U). Note that, by Lemma 2.2 below, if V is a multiply
connected Fatou component then there is an N N such that, for n N, V is a
n
∈ ≥
multiplyconnectedFatoucomponentwhichsatisfiesthehypothesesofTheorem1.1.
Using Theorem 1.1 we prove the following result.
Theorem 1.2. Suppose that f is a transcendental entire function and that L is
R
a fundamental loop of f. Then either L F(f) or L J(f).
R R
⊂ ⊂
AsecondconsequenceofTheorem1.1isthatwhenafundamentallooplieswithin
amultiplyconnectedFatoucomponent,U,itisoftenpossibletosaymoreaboutthe
nature of this set. In fact, there is a close relationship between some fundamental
loops of f and some level sets of the non-constant positive harmonic function h
that was introduced by Bergweiler, Rippon and Stallard in [8, Theorem 1.2], and
ON FUNDAMENTAL LOOPS AND THE FAST ESCAPING SET 3
usedto provemany geometricpropertiesofmultiply connectedFatoucomponents.
The function h is defined by
log fn(z)
(1.2) h(z)= lim | | , for z U, some z U.
n→∞log fn(z0) ∈ 0 ∈
| |
Our result is as follows.
Theorem 1.3. Suppose that f is a transcendental entire function. Then there
exists R′ = R′(f) > 0 such that the following holds. If U is a multiply connected
Fatou component of f, such that U surrounds the origin, dist(0,U)>R′, and h is
as defined as in (1.2), then
(a) if L U is a fundamental loop, then h(z) is constant on L and so L is an
R R R
⊂
analytic Jordan curve;
(b) if Γ is a level set of h, then Γ has a component γ which surrounds the origin
and there is a fundamental loop L such that L γ.
R R
⊂
It follows from these results that the fundamental loops of a transcendental
entire function can have very varied geometrical properties. For example, consider
the transcendental entire function f given in [8, Example 3]. This has a multiply
connected Fatou component U with the property that
max log z :z ∂ U
out n
lim { | | ∈ } = .
n→∞ min log z :z ∂outUn ∞
{ | | ∈ }
ByTheorem1.1,thereisafundamentalloopoff whichcoincideswith∂ U ,and
out n
so is far from circular for large values of n. However, there are also fundamental
loopsoff whichlieinsideU ,foreachn N. ByTheorem1.3(a)theseareanalytic
n
∈
Jordan curves, and by [8, Theorem 7.1] can be approximately circular.
A key tool in the proofs of these theorems is a function R , defined in (4.1)
A
below, which for a point z is the largest R such that z A (f). In general this
R
∈
functioncanonlybedefinedinasubsetofA(f). InSection6weshowthat,subject
to a certain normalisation,this definition can in fact be extended in a natural way
to the whole complex plane. We show that, in this case, there is an alternative
characterisation of A(f). We also show that the function R has a number of in-
A
teresting properties.
The structure of this paper is as follows. First, in Section 2, we state a num-
ber of results required in the proof of our main theorems. With the exception of
Lemma 2.6, these are all known results. In Section 3 we prove a new result, which
states that if a transcendental entire function has a certain property with respect
to a nested sequence of bounded simply connected domains, then there is a fixed
point which has a certain ‘attracting’ property. This may be of independent inter-
est. In Section 4 we show that the function R can be defined in certain multiply
A
connected Fatou components, and prove several preparatory lemmas. In Section 5
weproveTheorems1.1,1.2and1.3. Finally,inSection6westateandproveseveral
results regarding the case when R can be defined in the whole complex plane.
A
2. Background material
We use the following notation for an annulus and a disc
A(r ,r )= z :r < z <r , for 0<r <r ,
1 2 1 2 1 2
{ | | }
4 D.J.SIXSMITH
B(ζ, r)= z : z ζ <r , for 0<r.
{ | − | }
We require the following two well known facts about the maximum modulus of
a transcendental entire function:
(2.1) logM(et) is a convex and increasing function of t,
and
logM(r)
(2.2) as r .
logr →∞ →∞
We often use the following lemma [8, Theorem 2.2], generally with n=1.
Lemma2.1. Letf beatranscendentalentirefunction. ThenthereexistsR = R (f) > 0
0 0
such that, for all 0<c′ <1<c, and all n N,
∈
(2.3) M(rc,fn) M(r,fn)c, for r >R ,
0
≥
and
(2.4) M(rc′,fn) M(r,fn)c′, for r >R1/c′.
≤ 0
We denote the inverse function of M, when this is defined, by M−1. For sim-
plicity we write M−n, for n N, to denote n repeated iterations of M−1. Observe
∈
that M−1(r) is defined for r [f(0), ) and is strictly increasing. Moreover,
∈ | | ∞
by (2.1) and [16, Theorem 7.2.2],logM−1(es) is a concave and increasing function
of s. Also, if R is the constant from Lemma 2.1, then it follows from (2.3) that
0
(2.5) M−1(rc) M−1(r)c, for r >max M(R ), f(0) , c>1.
0
≤ { | |}
We next require a number of results concerning multiply connected Fatou com-
ponents. Our first is the following well-known result of Baker [1, Theorem 3.1],
which we often use without comment.
Lemma 2.2. Suppose that f is a transcendental entire function and that U is a
multiply connected Fatou component of f. Then each U is bounded and multiply
n
connected, U surrounds U for large n, and U as n .
n+1 n n
→∞ →∞
In particular,if U is a multiply connectedFatou component, then U is bounded
and so fn : U U is a proper map, for n N; see [15, Corollary 1]. Hence
n
U =fn(U), for→n N. ∈
n
∈
We need a number of results from[8]. Suppose that f is a transcendentalentire
function with a multiply connected Fatou component U = U , and let z U be
0 0
∈
fixed. Itfollowsfrom[8,Theorem1.2]thatthereexistsα>0suchthat,forlargen,
themaximumannuluscentredattheorigin,containedinU andcontainingfn(z )
n 0
is of the form
(2.6) B =A(ran,rbn), where r = fn(z ), 0<a <1 α<1+α<b .
n n n n | 0 | n − n
We require part of [8, Theorem 1.5].
Lemma 2.3. Suppose that f is a transcendental entire function with a multiply
connected Fatou component U, and let z U. For large n N, let r , a and b
0 n n n
∈ ∈
be as defined in (2.6), and let a denote the smallest value such that
n
a
z : z =r n ∂ U = .
n inn n
{ | | }∩ 6 ∅
Then, as n ,
→∞
a a [0,1), a a, and b b (1, ].
n → ∈ n → n → ∈ ∞
ON FUNDAMENTAL LOOPS AND THE FAST ESCAPING SET 5
Wealsoneedthefollowing[8,Theorem1.3]whichshowsthatanycompactsubset
of U eventually iterates into the maximal annulus B .
n
Lemma 2.4. Let f, U, z be as in Lemma 2.3. For large n N, let r , a , b
0 n n n
and B be as in (2.6). Then, for each compact set C U, ther∈e exists N N such
n
⊂ ∈
that
(2.7) fn(C) C B , for n N,
n n
⊂ ⊂ ≥
where
(2.8) C =A ran+2πδn,rbn(1−3πδn) , with δ =1/ logr .
n n n n n
Wealsoneedthefo(cid:16)llowing,whichshowst(cid:17)hatwithinC thpemodulusoff isvery
n
close to the maximum modulus, for large values of n. This is summarised from
[8, Theorem 5.1(b)].
Lemma 2.5. Let f, U and z be as in Lemma 2.3. For large n N, let r , a
0 n n
∈
and b be as in (2.6), and let δ = 1/√logr . Then, there exists N such that for
n n n
n N, and m N,
≥ ∈
(2.9) log fm(z) (1 δ )logM(z ,fm), for z A ran+2πδn,rbn−2πδn .
| |≥ − n | | ∈ n n
The following is a straightforwardconsequence of these(cid:0)lemmas. (cid:1)
Lemma 2.6. Let f and U be as in Lemma 2.3, let z U and let 0<c<1. Then
there exists N N such that ∈
∈
(2.10) fn+m(z) Mm(fn(z)c), for n N, m N.
| |≥ | | ≥ ∈
Proof. Fix z U, and let β =1 1/√logr , where r = fn(z ), for n N. It
0 n n n 0
∈ − | | ∈
is well-known that it follows from Lemma 2.2 that U A(f). Hence there exists
ℓ N such that r Mn(R), for n N, where R>⊂0 is such that Mn(R)
n+ℓ
as∈n . It follows≥from (2.2) that w∈e can choose N N sufficiently large t→hat∞
→∞ ∈
∞
(2.11) β >c.
k
k=N
Y
Now let z U. We can further assume that N is sufficiently large that
∈
fn(z)c >R , for n N,
0
| | ≥
where R is the constant from Lemma 2.1. Now, by Lemma 2.3,
0
C A ran+2πδn,rbn−2πδn ,
n ⊂ n n
for large values of n. Hence, we ca(cid:0)n assume, by Lemm(cid:1)a 2.4 and Lemma 2.5, that
N is sufficiently large that
(2.12) log fn+1(z) β logM(fn(z)), for n N.
n
| |≥ | | ≥
Hence, by (2.12) and (2.4),
(2.13) fn+1(z) M(fn(z))βn M(fn(z)βn), for n N.
| |≥ | | ≥ | | ≥
By repeated application of (2.13) and (2.4), and by (2.11), we have that
fn+m(z) Mm(fn(z)Qmk=−01βn+k) Mm(fn(z)c), for n N, m N,
| |≥ | | ≥ | | ≥ ∈
as required. (cid:3)
We also need the following [21, Theorem 2.3].
6 D.J.SIXSMITH
Lemma 2.7. Let f be a transcendental entire function and let η >1. There exists
R′ =R′(f)>0 such that if r >R′, then there exists
0 0 0
z′ A(r,ηr) A(f)
∈ ∩
with
fn(z′) >Mn(r,f), for n N,
| | ∈
and hence
z′ A (f) and M(ηr,fn)>Mn(r,f), for n N.
r
∈ ∈
Finally, in Section 6 we need the following well-known result [20, Lemma 2.1].
Lemma2.8. Letf beatranscendentalentirefunction, let K beacompact set with
K E(f) = and let ∆ be an open neighbourhood of z J(f). Then there exists
N ∩ N such t∅hat fn(∆) K, for n N. ∈
∈ ⊃ ≥
HereO−(z)= w :fn(w)=z, for some n N andE(f)= z :O−(z) is finite .
{ ∈ } { }
The set E(f) contains at most one point.
3. A map on a nested sequence of domains
In this section we prove a result about the existence and properties of a fixed
point for certain transcendental entire functions. This may be of independent in-
terest. For a hyperbolic domain V, we write [w,z] for the hyperbolic distance
V
between w and z in V. The main result of this section is as follows.
Theorem3.1. Supposethatf isatranscendentalentirefunction,andthat(G )
n n≥0
is a sequence of bounded simply connected domains such that
(3.1) G G and f(∂G )=∂G , for n=0,1,2, .
n n+1 n n+1
⊂ ···
Then there exists α G , a fixed point of f, such that, if K G is compact, then
0 0
∈ ⊂
[α,fn(z)] 0 as n , uniformly for z K.
Gn → →∞ ∈
To proveTheorem3.1we requirethe followinglemma. Define D= z : z <1 .
{ | | }
Lemma 3.1. Suppose that (B ) is a sequence of analytic functions from D to
n n≥0
D. Suppose also that there exist α D and λ (0,1) such that
∈ ∈
(3.2) B (α)=α and B′(α) λ, for n=0,1,2, .
n | n |≤ ···
Then, if K′ is a compact subset of D,
B B (z) α as n , uniformly for z K′.
n 0
◦···◦ → →∞ ∈
Proof. ByconjugatingwithaMo¨biusmapifnecessary,wemayassumethatα=0.
A result of Beardon and Carne [3, p.217] states that if g :D D is an analytic
→
function with g(0)=0, then
z + g′(0)
g(z) z | | | | , for z D,
| |≤| | 1+ g′(0)z ∈
(cid:18) | |(cid:19)
in which case
r+ g′(0)
M(r,g) r | | , for r (0,1).
≤ 1+ g′(0)r ∈
(cid:18) | | (cid:19)
ON FUNDAMENTAL LOOPS AND THE FAST ESCAPING SET 7
Now, (r+x)/(1+rx) is an increasing function of x, for r (0,1). Hence, by
∈
(3.2),
r+λ
(3.3) M(r,B ) r =µ(r)<r, for r (0,1), n=0,1,2, .
n
≤ 1+λr ∈ ···
(cid:18) (cid:19)
Note that µ(r) is a strictly increasing function of r (0,1). Let r (0,1) be
0
∈ ∈
such that z r , for z K′. Then, by (3.3),
0
| |≤ ∈
B B (z) µn(r ), for z K′.
n−1 0 0
| ◦···◦ |≤ ∈
Now µ(0)= 0, µ(1)=1, and 0< µ(r) <r, for r (0,1). Hence µn(r ) 0 as
0
n . This completes the proof of the lemma. ∈ → (cid:3)
→∞
We now prove Theorem 3.1.
Proof of Theorem 3.1. By (3.1), the triple (f,G ,G ) is a polynomial-like map in
0 1
the sense of Douady and Hubbard [10]. Hence, by [2, Lemma 3] (see also [12,
Lemma 3]), there exists a point α G such that f(α)=α.
0
For n = 0,1,2, , let φ : D ∈G be a Riemann map such that φ (0) = α.
n n n
··· →
Note that, since hyperbolic distance is preserved under conformal maps, we have
that
(3.4) [α,fn(z)]Gn =[0,φ−n1◦fn(z)]D, for z ∈G0.
HenDceefiintesufuffinccetisotnossBhow:Dthat[D0,bφy−n1◦fn(z)]D →0asn→∞,uniformlyforz ∈K.
n
→
B =φ−1 f φ , for n=0,1,2, .
n n+1◦ ◦ n ···
ThenB is a propermapsuchthat B (0)=0,forn=0,1,2, , andsois a finite
n n
···
Blaschke product
pn z a mk,n
(3.5) B (z)=c zqn − k,n , for z D,
n n
1 a z ∈
k=1(cid:18) − k,n (cid:19)
Y
where pn,qn,mk,n N, cn = 1, 0 < ak,n < 1, and ak,n = ak′,n implies that
∈ | | | |
k =k′, for n=0,1,2, , and k=1,2, ,p . (See, for example, [14, p.35].)
n
··· ···
We claim that there exists λ (0,1) such that
∈
(3.6) B′(0) λ, for n=0,1,2, .
| n |≤ ···
Suppose first that q 2 for some k N. Then B′(0)=0, for n=0,1,2, .
k ≥ ∈ n ···
Suppose, on the other hand, that q =1, for n=0,1,2, . Then
n
···
pn
(3.7) B′(0) = a mk,n <1, for n=0,1,2, .
| n | | k,n| ···
k=1
Y
For n = 0,1,2, , and k = 1,2, ,p , set α = φ (a ). Then f(α ) = α,
n k,n n k,n k,n
··· ···
andsoφ−1 (α )isazeroofB . Withoutlossofgeneralitywecanassumethat
n+1 k,n n+1
α =α =φ (a ), for n=0,1,2, , k =1,2, ,p .
k,n k,n+1 n+1 k,n+1 n
··· ···
Now, by (3.1), [α ,α] [α ,α] . Hence, once again since hyperbolic
k,n Gn ≥ k,n+1 Gn+1
distance is preserved under conformal maps, a a .
k,n+1 k,n
| |≤| |
Moreover, m and m are both equal to the multiplicity of the zero α
k,n k,n+1 k,n
of f(z) α, and so m =m . Thus, by (3.7),
k,n k,n+1
−
B′ (0) B′(0), for n=0,1,2, .
| n+1 |≤| n | ···
8 D.J.SIXSMITH
This establishes (3.6).
Let K G be compact. By Lemma 3.1, applied with K′ =φ−1(K), we obtain
⊂ 0 0
that
[0,Bn−1◦···◦B1◦B0◦φ−01(z)]D →0 as n→∞, uniformly for z ∈K.
The result follows by (3.4), since
B B B φ−1 =φ−1 fn.
n−1◦···◦ 1◦ 0◦ 0 n ◦
(cid:3)
4. The function R defined in a multiply connected Fatou component
A
The main role of this section is to introduce the function R , which plays a key
A
roleintheproofofTheorem1.1. Beforestatingandprovingasequenceoflemmas,
we outline how these results are used.
Suppose that U is a multiply connected Fatou component which surrounds the
origin. We show that if U is sufficiently far from the origin, then we can define a
real-valued function R which, for each z U, is the largest value of R such that
A
∈
z A (f); see (4.1) below. It turns out that this function has a close relationship
R
∈
to fundamental loops. Indeed, where defined, R is strictly less than R in H ,
A R
and is at least equal to R on L . We then prove that the function R has certain
R A
continuity properties, and shares level sets with the function h defined in (1.2).
These facts allow us to show that;
(a) on ∂ U, R is equal to its infimum in U;
int A
(b) on ∂ U, R is at least equal to its supremum in U;
out A
(c) R does not achieve a maximum or a minimum in U.
A
Because of the close relationship between the function R and the definition of
A
fundamental loops, properties (a), (b) and (c) above can then be used to prove
Theorem 1.1 parts (a), (b) and (c) respectively. Theorems 1.2 and 1.3 then follow
quickly.
We start with a simple lemma.
Lemma 4.1. Suppose that f is a transcendental entire function and that A (f) is
R
a spider’s web. Then f has a fixed point.
Proof. Suppose that H is a fundamental hole of f. By [21, Lemma 7.2] we have
R
that the triple (f,H ,f(H )) is a polynomial-like map. The result follows by
R R
[2, Lemma 3]. (cid:3)
The following lemma is central to our results.
Lemma 4.2. Suppose that f is transcendental entire function. Then there exists
R′ =R′(f)>0suchthatthefollowingholds. SupposethatU isamultiplyconnected
Fatou component of f, which surrounds the origin and satisfies dist(0, U) > R′.
Define G as the complementary component of U which contains the origin, for
n n
n=0,1,2, . Then
···
(a) G G , for n=0,1,2, ;
n n+1
⊂ ···
(b) f(∂G )=∂G , for n=0,1,2, ;
n n+1
···
(c) for all z U there exists R=R(z) such that z A (f).
R
∈ ∈
ON FUNDAMENTAL LOOPS AND THE FAST ESCAPING SET 9
Proof. Firstwe note [21, Theorem1.9(a)] that A (f) is a spider’s web. Hence, by
R
Lemma 4.1, f has a fixed point α.
Let V be a multiply connected Fatou component of f. By Lemma 2.2 there is
an N N such that fN(V) surrounds both the origin and α, and also fn+1(V)
∈
surrounds fn(V) for n N.
≥
Choose R > 0 such that Mn(R) as n . Then [21, Theorem 1.2(a)]
→ ∞ → ∞
there is an L N such that fL(V) A (f). Set M = max L,N , and let
R
∈ ⊂ { }
R′ = max z : z fM(V) .
{| | ∈ }
Suppose that U is any multiply connected Fatou component such that U sur-
roundstheoriginandsatisfiesdist(0,U)>R′. ThenU certainlysurroundsfM(V).
Since U surrounds α, then U = f(U) surrounds α by the argument principle.
1
Moreover,U cannot meet either fM+1(V) or U, since ∂U J(f). Hence, by the
1 1
⊂
maximumprinciple,U surroundsbothfM+1(V)andU. Inductively,U surrounds
1 k
both fM+k(V) and U , for k N. Parts (a) and (c) of the lemma follow from
k−1
∈
this fact and the choice of M.
Finally we establish part (b). Choose n = 0,1,2, . Since f(G ) is open and
n
···
connected,anditsboundaryisinJ(f),itcannotmeettheboundaryofG . Now,
n+1
α G , and so f(G ) G = . Hence ∂f(G ) must lie in G , and so
n n n+1 n n+1
∈ ∩ 6 ∅
f(∂G ) G .
n n+1
⊂
Moreover, f is a proper map on the Fatou component U , and so
n
f(∂G ) f(∂U )=∂U .
n n n+1
⊂
Thus ∂f(G )=∂G , as required. (cid:3)
n n+1
Suppose that U is a multiply connected Fatou component which surrounds the
origin, and that dist(0,U)>R′, where R′ is the constant from Lemma 4.2. Then,
by Lemma 4.2(c) and by the continuity of M, we may define
(4.1) R (z)=max R:z A (f) , for z U.
A R
{ ∈ } ∈
The function R has some strong continuity properties, and shares level sets with
A
the function h.
Lemma 4.3. Suppose that f and U are as in Theorem 1.1, and that R is as in
A
(4.1). Then R is upper semicontinuous in U and continuous in U. Moreover, if h
A
is as in (1.2), then there exists a continuous strictly increasing function φ:R R
→
such that
(4.2) R (z)=φ(h(z)), for z U.
A
∈
Proof. We first prove that R is upper semicontinuous in U. Suppose that z U
A
∈
and that ǫ > 0. By the definition of R , we have that z / A (f). Hence
A ∈ RA(z)+ǫ
there is anN N such that fN(z) <MN(R (z)+ǫ). By continuity, there exists
A
∈ | |
a δ >0 such that
fN(z′) <MN(R (z)+ǫ), for z′ B(z, δ).
A
| | ∈
Hence R (z′)<R (z)+ǫ, for all z′ U B(z, δ). This completes the proof that
A A
∈ ∩
R is upper semicontinuous in U.
A
ToprovethatR iscontinuousinU weneedtoprovethatR is lowersemicon-
A A
tinuous at z U. Suppose, to the contrary, that R is not lower semicontinuous
A
∈
10 D.J.SIXSMITH
at z. Then there exists ǫ > 0 such that the following holds. If ∆ U is a neigh-
⊂
bourhood of z, then there is a z′ ∆ such that R (z) ǫ > R (z′), in which
A A
∈ −
case z′ ∈/ ARA(z)−ǫ(f). There exists, therefore, a sequence (zk)k∈N of points of U,
distinct from but tending to z, and a sequence (nk)k∈N of integers such that
fnk(z ) <Mnk(R (z) ǫ), for k N.
k A
| | − ∈
Hence, for each k N,
∈
fnk(z ) <Mnk(R (z) ǫ)<Mnk(R (z)) fnk(z),
k A A
| | − ≤| |
which implies that
(4.3) logMnk(RA(z)) < log|fnk(z)| , for k N.
logMnk(RA(z) ǫ) log fnk(zk) ∈
− | |
We now establish a contradiction by showing that the right-hand side of (4.3) has
an upper bound which tends to 1 as k , but the left-hand side is greater than
→∞
somec>1,forsufficientlylargevaluesofk. Notethatwecanassumethatn
k
→∞
as k .
→∞
We mayassumethatǫ issufficiently smallthatMn(R (z) ǫ) asn .
A
− →∞ →∞
Hence we can choose N large enough that MN(R (z) ǫ)> R , where R is the
A 0 0
−
constant from Lemma 2.1. Set
logMN(R (z))
r =MN(R (z) ǫ) and c= A >1.
A
− logr
It follows by repeated application of (2.3) that we have
logMm(rc) clogMm(r), for m N.
≥ ∈
Hence
logMm+N(R (z)) logMm(rc)
A = c>1, for m N.
logMm+N(R (z) ǫ) logMm(r) ≥ ∈
A
−
This establishes our claim regarding the left-hand side of (4.3).
To establish our claim regarding the right-hand side of (4.3) we use some tech-
niques from[19], though we give the full details for completeness. For a hyperbolic
domain V, we let ρ denote the density of the hyperbolic metric in V.
V
Choose any w ,w J(f) with w =w , and put G=C w ,w . Note that
1 2 1 2 1 2
∈ 6 \{ }
[z,zk]U ≥[fnk(z),fnk(zk)]fnk(U) ≥[fnk(z),fnk(zk)]G = ρG(z)|dz|, for k ∈N,
ZΓk
where Γk is a hyperbolic geodesic in G joining fnk(z) to fnk(zk). By, for example,
[13, Theorem 9.14], there exist R>2 and C >0 such that
C
ρ (z) , for z R.
G
≥ z log z | |≥
| | | |
Choose K sufficiently large such that
fnk(z) > fnk(z ) >2R, for k K.
k
| | | | ≥
We then have
|fnk(z)| dr log fnk(z)
(4.4) [z,z ] ρ (z) dz C =Clog | | .
k U ≥ZΓk G | |≥ Z|fnk(zk)| rlogr (cid:18)log|fnk(zk)|(cid:19)