Maximal clus e se s along a bi a y cu es
L. Be nal-Gonz´alez, M.C. Calde ´on-Mo eno and J.A. P ado-Bassas∗
Abs ac
The exis ence o a dense linea mani old o holomo phic unc ions on a
Jo dan domain ha ing excep o ze o maximal clus e se along any cu e
ending o he bounda y wi h non o al oscilla ion alue se is shown.
Key wo ds and ph ases: clus e se , cu e ending o he bounda y, Jo dan
domain, dense linea mani old.
2000 Ma hema ics Subjec Classi ica ion: P ima y 30D40. Seconda y 30E10,
30H05.
1 In oduc ion and no a ion
Th oughou his pape we will use he ollowing s anda d no a ions: Nis he se
o posi i e in ege s, Cis he complex plane, D:= {z∈C:|z|<1}is he open uni
disk, B(a, ) (B(a, )) is he euclidean open (closed, esp.) ball wi h cen e a∈C
and adius > 0. Mo eo e , i Gis a domain (:= connec ed, nonemp y open subse )
o C, hen H(G) will s and o he space o holomo phic unc ions on G. I becomes
a comple ely me izable space (hence a Bai e space) when i is endowed wi h he
compac open opology (see [11, pages 238–239]). Finally, i Ais a subse o C hen
Adeno es i s closu e in Cwhile ∂A deno es i s bounda y in he ex ended complex
plane C∞:= C∪ {∞}. In pa icula , Twill s and o he uni ci cle ∂D.
A well-known in e pola ion heo em due o Weie s ass (see [13, Chap e 15])
asse s ha i a domain G⊂C, a sequence {an}∞
n=1 ⊂Gwi h no limi poin s in G
– ha is, ending o he bounda y– and a sequence {wn}∞
n=1 ⊂Ca e p esc ibed, hen
he e exis s a unc ion ∈H(G) such ha (an) = wn o all n∈N. In pa icula ,
∗This wo k is suppo ed in pa by he Plan Andaluz de In es igaci´on de la Jun a de Andaluc´ıa
FQM-127.
1
i we choose as {wn}∞
n=1 an enume a ion o he complex numbe s ha ing a ional eal
and imagina y pa s hen a unc ion ∈H(G) wi h { (an) : n∈N}dense in Cis
ob ained. Since a dense se wi h fini ely many poin s dele ed con inues o be a dense
se , we ge a unc ion ∈H(G) wi h maximal clus e se along he se {an}∞
n=1, in he
sense exp essed in he ollowing pa ag aph. No e also ha , equi alen ly, he densi y
o (A) o some ∈H(G) can be achie ed o e e y fixed non ela i ely compac
subse Ao G.
Assume ha Gis a domain in C, ha F:G→Cis a unc ion defined on Gand
ha Ais a subse o G. The clus e se o Falong Ais defined as he se
CA(F) = {w∈C: he e exis s a sequence {zn}∞
n=1 ⊂A ending o
some poin o ∂G such ha limn→∞ F(zn) = w}.
I is clea ha CA(F) is always closed and ha i CA(F)=∅ hen Ais no ela i ely
compac in G. The eade is e e ed o [5] and [12] o su eys o esul s abou clus e
se s. I 0∈∂G hen he clus e se o Falong Aa 0is defined as
CA(F, 0) = {w∈C: he e exis s a sequence {zn}∞
n=1 ⊂A ending o 0
such ha limn→∞ F(zn) = w}.
Again, CA(F, 0) is always closed. In addi ion, CA(F) = ∪
∈∂G
CA(F, ). I A=G
hen he subsc ip “A” is o en dele ed and he exp ession “along A” is d opped. An
impo an special case is he adial clus e se a 0, which is defined as Cϱ(F, 0) :=
CA(F) = CA(F, 0), whe e Ais he adius A={u 0:u∈[0,1)}.
I is an in e es ing p oblem o ob ain holomo phic unc ions wi h maximal clus e
se s, ha is, wi h clus e se s equal o C. In [1] i is shown ha he unc ions
∈H(G) ha ing maximal clus e se a e e y bounda y poin o m a esidual subse
(i.e. i s complemen is o fi s ca ego y) in H(G), while in [2] i is p o ed ha o a
p esc ibed non ela i ely compac subse A⊂G he se { ∈H(G) : (A) = C}is
esidual in H(G), om which i is easy o conclude ha o Aas be o e he e exis s a
esidual subse o H(G) all o whose unc ions ha e maximal clus e se along A. An
impo an special ins ance is ha o a cu e in G ending o he bounda y, ha is,
a con inuous map γ: [0,1) →Gsuch ha limu→1−γ(u) = ω:= he infini y poin o
he one-poin compac ifica ion o Go , equi alen ly, such ha o each compac se
K⊂G he e is u0=u0(K)∈[0,1) wi h γ(u)∈G K o all u > u0(in pa icula i
G=D hen γ ends o he bounda y i and only i limu→1−|γ(u)|= 1). By abuse o
lenguage we some imes iden i y γ=γ([0,1)). F om he abo e-men ioned esul o [2]
and om he ac ha a coun able in e sec ion o esidual subse s is again esidual
(so dense) one can ex ac ha i Γ is a gi en coun able amily o cu es in G ending
2
o he bounda y hen he e is a dense subse M⊂H(G) such ha Cγ( ) is maximal
o all ∈Mand all γ∈Γ.
In his pape we ob ain ha a leas o each Jo dan domain he e exis s a dense
linea mani old o holomo phic unc ions ha ing –excep o ze o– maximal clus e se
along any cu e ending o he bounda y wi h non o al oscilla ion alue se . Hence
we can say ha he se o unc ions wi h such app oxima ion p ope y is la ge no
only opologically bu also algeb aically.
2 The main esul
By a Jo dan domain we mean a domain in Cwhose bounda y in C∞is a opological
image o he uni ci cle T. I G⊂Cis a domain and A⊂Gis non ela i ely compac
hen i s oscilla ion alue se is he (nonemp y) se
Osc (A) = { ∈∂G : he e exis s a sequence {zn}∞
n=1 ⊂A
wi h limn→∞ zn= }.
We a e now eady o s a e ou main esul .
Theo em 2.1. Le Gbe a Jo dan domain. Then he e is a dense linea mani old
Din H(G)such ha o e e y ∈ D {0}and e e y cu e γ⊂G ending o he
bounda y wi h Osc (γ)=∂G we ha e Cγ( ) = C. In pa icula , (γ)is dense in C
o each pai ,γas be o e.
P oo . By he Osgood-Ca a h´eodo y heo em (see [9]) he e exis s an homeomo -
phism φ om he C∞-closu e o Gon o Dwhose es ic ion on Gis a holomo phic
isomo phism om Gon o D. Then i Dwe e he dense linea mani old ob ained o
H(D) hen he se D1:= { ◦φ: ∈ D} would be he desi ed linea mani old in
H(G). The de ails a e many bu easy, and hey a e le o he eade .
Hence we may suppose ha G=D om now on. Assume ha {P∗
n}∞
n=1 is a
coun able dense subse o H(D) ( o ins ance, an enume a ion o he holomo phic
polynomials ha ing coefficien s wi h a ional eal and imagina y pa s). Then we
conside a sequence {Pn}∞
n=1 whe e each P∗
noccu s infini ely many imes. We also
fix wo sequences { n},{sn}o posi i e eal numbe s sa is ying 1< s1< 2< s2<
· · · < n< sn<· · · and limn→∞ n= 1 = limn→∞ sn. Le us di ide Nin o infini ely
many s ic ly inc easing sequences {p(n, j) : j= 1,2, . . .}(n∈N). Fo fixed n∈N
3
we conside he se Fn⊂Dgi en by he disjoin union
Fn=B(0,n
n+ 1)∪
∞
∪
j=J(n)
Kj,
whe e J(n) := min{j∈N: j>n
n+1}and each Kjis he spi al compac se
Kj={( j+sj− j
4πθ) exp(iθ) : θ∈[0,4π]}.
Obse e ha each Kjhas connec ed complemen and ha he sequence {Kj}∞
j=1 goes
o T. No e also ha e e y Fnis closed in D. By D∞we will deno e he one-poin
compac ifica ion o D, whe eas ωwill s and o i s infini y poin . A simple glance
e eals ha D∞ Fnis connec ed (indeed, D Fnis connec ed and D Fn⊂D∞ Fn⊂
he closu e in D∞o D Fn) and locally connec ed a ω(by a simila eason). In
addi ion, Fnsa isfies he ollowing p ope y: Fo e e y compac subse K⊂D he e
exis s a neighbou hood Vo ωin D∞such ha no componen o he in e io F0
no
Fnin e sec s bo h Kand V; indeed, F0
n=B(0,n
n+1 ) and o any Kwe can choose
V:= {ω} ∪ { n
n+1 <|z|<1}. Unde hese h ee opological condi ions he Ne sesjan
heo em (see [7]) asse s he exis ence o a unc ion n∈H(D) app oaching a gi en
con inuous unc ion gn:Fn→Cwi h gnholomo phic in F0
nwi hin a p esc ibed e o
unc ion (= con inuous posi i e unc ion on Fn)ε(z). I we selec ε(z) := 1−|z|
n hen
we ob ain
| n(z)−gn(z)|<1− |z|
n(z∈Fn),(1)
whe e gn:Fn→Cis he unc ion defined as
gn(z) = 


Pn(z) i z∈B(0,n
n+1)
qji z∈Kp(n,j)and p(n, j)≥J(n)
0 i z∈Kp(k,j)(k=n) and p(k, j)≥J(n).
We ha e deno ed he e by {qj}∞
j=1 any fixed dense sequence in C. Obse e ha ,
i ially, gnis con inuous on Fnand holomo phic in F0
n, so Ne sesjan’s heo em applies
p ope ly.
Le us define Das he linea span
D= span { n:n∈N}.
O cou se, Dis a linea submani old o H(D), and Dis dense because { n}∞
n=1 is.
Indeed, om (1) we ha e ha
| n(z)−Pn(z)|<1
n o all z∈B(0,n
n+ 1).
4
Then i we fix a unc ion P∗
m he e exis s a sequence n1< n2<· · · wi h Pnj=P∗
m o
all j∈N. Now i K⊂Dis compac hen he e is j0∈Nsuch ha K⊂B(0,nj
nj+1 )
o e e y j > j0. The e o e
| nj(z)−P∗
m(z)|<1
nj
o all z∈Kand all j > j0,
so nj→P∗
m(j→ ∞) uni o mly on compac a in H(D). Hence he closu e o { n:
n∈N}in H(D) con ains he dense se {P∗
m:m∈N}, which p o es he densi y o
{ n}∞
n=1.
I emains o show ha o e e y p esc ibed cu e γ∈Gas in he hypo hesis and
o e e y unc ion ∈ D {0}we ha e Cγ( ) = C. No e ha o such unc ion he e
exis N∈Nand complex scala s λ1, . . . , λNsuch ha λN= 0 and =λ1 1+· · · +
λN N. Since Osc (γ)=Tand γshould escape owa ds T, his cu e mus in e sec
all spi als Kjexcep fini ely many o hem; indeed, i his we e no he case hen
he shape o Kj’s oge he wi h he con inui y o γwould o ce γ o make infini ely
many windings a ound he o igin while app oaching T, which would con adic he
hypo hesis Osc (γ)=T. The e o e he e exis s j0∈Nsuch ha p(k, j0)≥J(N)
(k= 1, . . . , N) and γ∩Kp(N,j)=∅(j≥j0). Choose poin s zj∈γ∩Kp(N,j)(j≥j0).
Then by (1) we ob ain, o e e y j≥j0,
| N(zj)−qj|=| N(zj)−gN(zj)|<1− |zj|
N≤1− |zj| ≤ 1− j
and
| n(zj)|=| n(zj)−gn(zj)|<1− |zj|
n≤1− j(n= 1, . . . , N −1).
Hence we ge
| (zj)−λNqj|=|λ1 1(zj) + · · · +λN N(zj)−λNqj|
≤ |λN| · | N(zj)−qj|+
N−1
∑
n=1
|λn n(zj)|
<(N
∑
n=1
|λn|)(1 − j)→0 (j→ ∞).
Bu since λN= 0 he sequence {λNqj:j∈N}is dense in C, so o gi en α∈C
he e is a sequence {j1< j2<· · ·} ⊂ Nwi h λNqjk→αas k→ ∞. Now we can
selec a sequence {k(1) < k(2) <· · ·} ⊂ Nand a poin ∈Twi h wl:= zjk(l)→
(l→ ∞). Then {wl}∞
l=1 ⊂γand (wl) = (wl)−λNqjk(l)+λNqjk(l)→α(l→ ∞), so
α∈Cγ( ). In o he wo ds, Cγ( ) = C, as equi ed.
5

In iew o Theo em 2.1, wo na u al ques ions a ise, namely:
(a) Is i possible o eplace he a bi a y cu e γ o an a bi a y sequence {zn}∞
n=1
ending o he bounda y (e en wi h Osc ({zn}∞
n=1)=∂G)? The elemen a y
P oposi ion 2.2 below answe s his ques ion in he nega i e.
(b) I is clea ha a simila esul o Theo em 2.1 alls down i one desi es ha
belongs o a subspace o bounded unc ions. Bu e en wi hou his boundedness
es ic ion he s a emen may be alse. Fo ins ance, i is in he Ha dy space
Hp(see below) o he uni disk hen Fa ou’s heo em asse s ha he adial
limi lim →1− ( eiθ) exis s and is fini e o all θ∈A, whe e A=A is a subse
o [0,2π] such ha he Lebesgue measu e o [0,2π] Ais ze o, see [6]. The e o e
Cγ( ) is a single on o each adial cu e γ={ eiθ : ∈[0,1)}(θ∈A).
Ne e heless, making a link o a mo i a ing esul men ioned in Sec ion 1, we
could ask whe he a leas o a p esc ibed coun able amily o cu es in D
ending o T he asse ion o Theo em 2.1 holds in Hp. Theo em 2.5 below will
p o ide his ime a posi i e answe , e en wi hou he es ic ion Osc (γ)=T.
P oposi ion 2.2. I G⊂Cis a bounded domain and ∈H(G) hen he e a e a
poin ∈∂G, a alue A∈Cand a sequence {zn}∞
n=1 ⊂G ending o such ha
limn→∞ (zn) = A.
P oo . I has infini ely many ze os hen he esul ollows om he Analy ic Con-
inua ion P inciple. Suppose now ha has fini ely many ze os. Define g= /P,
whe e P≡1 i has no ze os whe eas P(z)≡(z−a1)· · · (z−ap) i a1, . . . , apa e
he ze os o , coun ing acco ding hei mul iplici ies. Then gis in H(G) and has
no ze os. Le us fix a sequence {Kn}∞
n=1 o compac subse s o Gwhich is exhaus-
i e, in he sense ha i s union is Gand Kn⊂K0
n+1 (n∈N). Wi hou loss o
gene ali y, we can suppose K0
1=∅. Choose any poin a∈K0
1, so a∈K0
n o all
n. Since ghas no ze os, he Minimum Modulus P inciple ells us ha he minimum
o |g|on Knis a ained a some poin an∈∂Kn, he e o e |g(an)| ≤ |g(a)|. Then
| (an)|=|P(an)| · |g(an)| ≤ M:= |g(a)| · supz∈G|P(z)|(n∈N), whe e Mis fini e
because Gis bounded. Summa izing, we ha e ob ained a sequence {an}∞
n=1 ⊂G
such ha { (an)}∞
n=1 is bounded. Bu he exhaus i i y p ope y o {Kn}∞
n=1 im-
plies ha o a gi en compac se K⊂G he e is n0∈Nwi h K⊂Kn0, so
{an:n>n0} ∩ K=∅, whence he compac ness o Gleads us up o a poin ∈∂G
wi h bn→ (n→ ∞) o some subsequence {bn}∞
n=1 o {an}∞
n=1. Finally, he bound-
edness o { (bn)}∞
n=1 gua an ees ha (zn)→A(n→ ∞) o some A∈Cand some
subsequence {zn}∞
n=1 o {bn}∞
n=1.
6
Recall ha a sequence Tn:X→Y(n∈N) o con inuous linea mappings be ween
wo opological ec o spaces X,Yis called uni e sal o hype cyclic whene e he e
exis s a ec o x∈X–called uni e sal o {Tn}∞
n=1– whose o bi {Tnx:n∈N}is
dense in Y. By U({Tn}) we will deno e he se o such uni e sal ec o s. I his se
is dense in Y hen we say ha {Tn}∞
n=1 is densely uni e sal. See [8] o an excellen
su ey (upda ed ill 1999) abou concep s, his o y and esul s ela ed o his opic.
The ollowing auxilia y esul can be ound in [3, Theo em 3.1].
Lemma 2.3. Assume ha X,Ya e me izable opological ec o spaces and ha X
is Bai e and sepa able. Suppose ha , o each k∈N,T(k)
n:X→Y(n∈N)is a
sequence o con inuous linea mappings be ween Xand Y. Assume ha o e e y k
and e e y sequence {n1< n2<· · ·} ⊂ N he sequence {T(k)
nj}∞
j=1 is densely uni e sal.
Then he e exis s a dense linea mani old M⊂Xsuch ha
M {0} ⊂ ∩
k∈N
U({T(k)
n}).
I 0 < p < ∞ hen he Ha dy space Hpis he class o unc ions ∈H(D) o which
∥ ∥p:= sup
0< <1
(∫2π
0
| ( eiθ)|pdθ
2π)1/p <∞. I becomes a Banach space o 1 ≤p < ∞
when endowed wi h he no m ∥ ∥p. In he nine ies P. Bou don and J.H. Shapi o
we e able o p o e ha o p= 2 he e is a esidual subse o unc ions ∈Hp o
which he o bi { ◦ψn:n∈N}is dense in Hp, whe e ψnis he n h-i e a e o an
au omo phism ψo Dwi hou fixed poin s in D(see [4] and [14, Chap e 7], whe e
many esul s o his kind can be ound). Thei p oo equally wo ks o 1 ≤p < ∞
because i is ul ima ely based on he ac s ha excep o pe haps one poin o T
he sequence ψn(z) ends o a cons an alue α∈Tand ha o e e y β∈ T he
collec ion o polynomials anishing a βis dense in Hp, which in u n is a consequence
o Beu ling’s app oxima ion heo em, see [6, pages 113–114]. Now we deno e by φa
(a∈D) he au omo phism o Dgi en by φa(z) = z+a
1+az . I is a s aigh o wa d exe cise
o check ha i {an}∞
n=1 ⊂Dand an→α∈T hen φan( )→α(n→ ∞) o e e y
∈T {−α}. Wi h hese hin s he in e es ed eade will find no difficul y in p o ing
he ollowing ex ension o Bou don-Shapi o’s esul .
Lemma 2.4. Le be p esc ibed a numbe p∈[1,∞)and a sequence {an}∞
n=1 ⊂D
ending o a bounda y poin . Then he unc ions ∈Hp o which he o bi { ◦φan:
n∈N}is dense in Hp o m a esidual subse .
Wi h he help o he la e wo lemmas we can conclude his sec ion by p o ing
he ollowing heo em. We ema k ha since Hp-con e gence is s onge han local
uni o m con e gence, he mani old Dob ained below becomes dense also in H(D).
7
Theo em 2.5. Suppose ha p∈[1,∞)and ha Γis a coun able collec ion o cu es
in D ending o he bounda y. Then he e is a dense linea mani old Din Hpsuch
ha Cγ( ) = C o e e y ∈ D {0}and e e y γ∈Γ.
P oo . Since Γ is coun able, we can w i e Γ = {γk:k∈N}whe e each γkis a cu e
in D ending o T, whence o e e y kwe can pick a sequence {a(k)
n:n∈N} ⊂ γk
ending o some poin αk∈T. I {n1< n2<· · ·} ⊂ N hen we ha e also ha
a(k)
nj→αkas j→ ∞. Thus by Lemma 2.4 he unc ions ∈Hp o which he o bi
{ ◦φa(k)
nj
:j∈N}is dense in Hp o m a esidual (so dense) subse o Hp o e e y
k∈N. In o he wo ds, each sequence {T(k)
nj}∞
j=1 (k∈N) is densely uni e sal, whe e
T(k)
ndeno es he composi ion ope a o ∈Hp7→ ◦φa(k)
n∈Hp. Bu X:= Hp=: Y
is a Bai e me izable sepa able opological ec o space, hence Lemma 2.3 yields he
exis ence o a dense linea mani old D ⊂ Hpsuch ha D {0} ⊂ ∩k∈NU({T(k)
n}).
Finally, ake a unc ion ∈ D {0}and a cu e γ=γk∈Γ. Then ∈ U({T(k)
n}),
which implies ha { ◦φa(k)
n:n∈N}is dense in Hp, so in H(D). In pa icula , he
se {( ◦φa(k)
n)(0) : n∈N}={ (a(k)
n) : n∈N}is dense in {g(0) : g∈H(D)}=C.
Bu {a(k)
n:n∈N} ⊂ γand a(k)
n→αk∈T, so C⊂Cγ( ) and we a e done.
3 Final ema ks
1. R. Ten hoff has ecen ly cons uc ed (see [15, Kapi el 3]) a dense se o unc ions
∈H(D) sa is ying he ollowing p ope y: Fo e e y 0∈T, e e y compac
subse K⊂Dwi h connec ed complemen and e e y con inuous unc ion g:
K→Cwi h g∈H(K0), he e exis s a sequence o unc ions n:K→
{u 0:u∈[0,1)}–no necessa ily holomo phic no con inuous– such ha
limn→∞ n(z) = z0 o all z∈Kand ◦ n→guni o mly on K. I we choose
specially K={0} hen i is de i ed he ollowing pa icula case o Theo em
2.1: The e is a dense se o unc ions ∈H(D) all o whose adial clus e se s
Cϱ( , 0) a e maximal.
2. In connec ion wi h he las ema k he ollowing ques ion a ises: Is he se
{ ∈H(D) : Cϱ( , 0) = C o all 0∈T} esidual in H(D)? We do no
know he answe , bu we a e able a leas o show he nex esul : The se
{ ∈H(D) : Cϱ( , 0) = C o all 0belonging o some esidual se A=A ⊂T}
is esidual in H(D). Indeed, by [1] he unc ions ∈H(D) wi h maximal clus e
se C( , 0) a any 0∈Tis esidual, and by Collingwood’s maximali y heo em
8
(see [5, Theo em 4.8]) i F:D→Cis con inuous, γis a cu e in D e mina ing
a 1 (in pa icula , γcan be he adius [0,1)) and γ := ·γ( ∈T) hen
Cγ (F, ) = C(F, ) on a esidual se (depending on F) o poin s on T.
3. P oposi ion 2.2 showed ha a leas o a bounded domain G⊂C, he e is no
unc ion in H(G) wi h maximal clus e se along any sequence {zn}∞
n=1 ⊂G
ending o he bounda y ∂G. Howe e , i we d op he amoun o sequences
{zn}∞
n=1 hen i is possible o ge a posi i e esul . Gi en A⊂C, we deno e by
A′ he se o i s accumula ion poin s in C∞.
P oposi ion 3.1. Le Abe a non ela i ely compac subse o a domain G⊂C.
Then he se
M:= { ∈H(G) : CA( , ) = C o all ∈A′∩∂G}
is esidual in H(G).
P oo . Le { k}∞
k=1 be a coun able dense subse o A′∩∂G. Fo each k, we
choose a sequence {a(k)
n}∞
n=1 ⊂Awi h a(k)
n→ k(n→ ∞). By [2], i is known
ha he se s { ∈H(G) : C{a(k)
n:n∈N}( , k) = C}(k∈N) a e esidual, hence
by Bai e’s heo em
D:= ∩
k∈N
{ ∈H(G) : C{a(k)
n:n∈N}( , k) = C}
is esidual.
Le ∈ D and ∈A′∩∂G. I we p o e ha CA( , ) = C, hen we would ha e
∈ M. Thus D ⊂ M and Mwould be esidual.
Le {wn}∞
n=1 be a coun able dense subse o C. By induc ion, we can cons uc
an inc easing sequence {mn}∞
n=1 ⊂Nsuch ha
| (a(k)
mn)−wn|<1
n(k= 1, . . . , n;n∈N).(2)
Fix a alue w∈C. The e is an inc easing sequence {in}∞
n=1 ⊂Nwi h
|win−w|<1
in
(n∈N).(3)
The poin is an accumula ion poin o he se
{a(k)
min:k= 1, . . . , in;n∈N}
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