Maximal cluster sets along arbitrary curves

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} 9