Maximal cluster sets along arbitrary curves

Maximal cluster sets along arbitrary curves L. Bernal-Gonz´alez, M.C. Calder´on-Moreno and J.A. Prado-Bassas∗ Abstract The existence of a dense linear manifold of holomorphic functions on a Jordan domain having except for zero maximal cluster set along any curve tending to the boundary with nontotal oscillation value set is shown. Key words and phrases: cluster set, curve tending to the boundary, Jordan domain, dense linear manifold. 2000 Mathematics Subject Classification: Primary 30D40. Secondary 30E10, 30H05. 1 Introduction and notation Throughout this paper we will use the following standard notations: Nis the set of positive integers, Cis the complex plane, D:= {z∈C:|z|<1}is the open unit disk, B(a, r) (B(a, r)) is the euclidean open (closed, resp.) ball with center a∈C and radius r > 0. Moreover, if Gis a domain (:= connected, nonempty open subset) of C, then H(G) will stand for the space of holomorphic functions on G. It becomes a completely metrizable space (hence a Baire space) when it is endowed with the compact open topology (see [11, pages 238–239]). Finally, if Ais a subset of Cthen Adenotes its closure in Cwhile ∂A denotes its boundary in the extended complex plane C∞:= C∪ {∞}. In particular, Twill stand for the unit circle ∂D. A well-known interpolation theorem due to Weierstrass (see [13, Chapter 15]) asserts that if a domain G⊂C, a sequence {an}∞ n=1 ⊂Gwith no limit points in G –that is, tending to the boundary– and a sequence {wn}∞ n=1 ⊂Care prescribed, then there exists a function f∈H(G) such that f(an) = wnfor all n∈N. In particular, ∗This work is supported in part by the Plan Andaluz de Investigaci´on de la Junta de Andaluc´ıa FQM-127. 1 if we choose as {wn}∞ n=1 an enumeration of the complex numbers having rational real and imaginary parts then a function f∈H(G) with {f(an) : n∈N}dense in Cis obtained. Since a dense set with finitely many points deleted continues to be a dense set, we get a function f∈H(G) with maximal cluster set along the set {an}∞ n=1, in the sense expressed in the following paragraph. Note also that, equivalently, the density of f(A) for some f∈H(G) can be achieved for every fixed nonrelatively compact subset Aof G. Assume that Gis a domain in C, that F:G→Cis a function defined on Gand that Ais a subset of G. The cluster set of Falong Ais defined as the set CA(F) = {w∈C: there exists a sequence {zn}∞ n=1 ⊂Atending to some point of ∂G such that limn→∞ F(zn) = w}. It is clear that CA(F) is always closed and that if CA(F)=∅then Ais not relatively compact in G. The reader is referred to [5] and [12] for surveys of results about cluster sets. If t0∈∂G then the cluster set of Falong Aat t0is defined as CA(F, t0) = {w∈C: there exists a sequence {zn}∞ n=1 ⊂Atending to t0 such that limn→∞ F(zn) = w}. Again, CA(F, t0) is always closed. In addition, CA(F) = ∪ t∈∂G CA(F, t). If A=G then the subscript “A” is often deleted and the expression “along A” is dropped. An important special case is the radial cluster set at t0, which is defined as Cϱ(F, t0) := CA(F) = CA(F, t0), where Ais the radius A={u t0:u∈[0,1)}. It is an interesting problem to obtain holomorphic functions with maximal cluster sets, that is, with cluster sets equal to C. In [1] it is shown that the functions f∈H(G) having maximal cluster set at every boundary point form a residual subset (i.e. its complement is of first category) in H(G), while in [2] it is proved that for a prescribed nonrelatively compact subset A⊂Gthe set {f∈H(G) : f(A) = C}is residual in H(G), from which it is easy to conclude that for Aas before there exists a residual subset of H(G) all of whose functions have maximal cluster set along A. An important special instance is that of a curve in Gtending to the boundary, that is, a continuous map γ: [0,1) →Gsuch that limu→1−γ(u) = ω:= the infinity point of the one-point compactification of Gor, equivalently, such that for each compact set K⊂Gthere is u0=u0(K)∈[0,1) with γ(u)∈G\Kfor all u > u0(in particular if G=Dthen γtends to the boundary if and only if limu→1−|γ(u)|= 1). By abuse of lenguage we sometimes identify γ=γ([0,1)). From the above-mentioned result of [2] and from the fact that a countable intersection of residual subsets is again residual (so dense) one can extract that if Γ is a given countable family of curves in Gtending 2 to the boundary then there is a dense subset M⊂H(G) such that Cγ(f) is maximal for all f∈Mand all γ∈Γ. In this paper we obtain that at least for each Jordan domain there exists a dense linear manifold of holomorphic functions having –except for zero– maximal cluster set along any curve tending to the boundary with nontotal oscillation value set. Hence we can say that the set of functions with such approximation property is large not only topologically but also algebraically. 2 The main result By a Jordan domain we mean a domain in Cwhose boundary in C∞is a topological image of the unit circle T. If G⊂Cis a domain and A⊂Gis nonrelatively compact then its oscillation value set is the (nonempty) set Osc (A) = {t∈∂G : there exists a sequence {zn}∞ n=1 ⊂A with limn→∞ zn=t}. We are now ready to state our main result. Theorem 2.1. Let Gbe a Jordan domain. Then there is a dense linear manifold Din H(G)such that for every f∈ D \ {0}and every curve γ⊂Gtending to the boundary with Osc (γ)=∂G we have Cγ(f) = C. In particular, f(γ)is dense in C for each pair f,γas before. Proof. By the Osgood-Carath´eodory theorem (see [9]) there exists an homeomorphism φfrom the C∞-closure of Gonto Dwhose restriction on Gis a holomorphic isomorphism from Gonto D. Then if Dwere the dense linear manifold obtained for H(D) then the set D1:= {f◦φ:f∈ D} would be the desired linear manifold in H(G). The details are many but easy, and they are left to the reader. Hence we may suppose that G=Dfrom now on. Assume that {P∗ n}∞ n=1 is a countable dense subset of H(D) (for instance, an enumeration of the holomorphic polynomials having coefficients with rational real and imaginary parts). Then we consider a sequence {Pn}∞ n=1 where each P∗ noccurs infinitely many times. We also fix two sequences {rn},{sn}of positive real numbers satisfying r1< s1< r2< s2< · · · < rn< sn<· · · and limn→∞ rn= 1 = limn→∞ sn. Let us divide Ninto infinitely many strictly increasing sequences {p(n, j) : j= 1,2, . . .}(n∈N). For fixed n∈N 3 we consider the set Fn⊂Dgiven by the disjoint union Fn=B(0,n n+ 1)∪ ∞ ∪ j=J(n) Kj, where J(n) := min{j∈N:rj>n n+1}and each Kjis the spiral compact set Kj={(rj+sj−rj 4πθ) exp(iθ) : θ∈[0,4π]}. Observe that each Kjhas connected complement and that the sequence {Kj}∞ j=1 goes to T. Note also that every Fnis closed in D. By D∞we will denote the one-point compactification of D, whereas ωwill stand for its infinity point. A simple glance reveals that D∞\Fnis connected (indeed, D\Fnis connected and D\Fn⊂D∞\Fn⊂ the closure in D∞of D\Fn) and locally connected at ω(by a similar reason). In addition, Fnsatisfies the following property: For every compact subset K⊂Dthere exists a neighbourhood Vof ωin D∞such that no component of the interior F0 nof Fnintersects both Kand V; indeed, F0 n=B(0,n n+1 ) and for any Kwe can choose V:= {ω} ∪ { n n+1 <|z|<1}. Under these three topological conditions the Nersesjan theorem (see [7]) asserts the existence of a function fn∈H(D) approaching a given continuous function gn:Fn→Cwith gnholomorphic in F0 nwithin a prescribed error function (= continuous positive function on Fn)ε(z). If we select ε(z) := 1−|z| nthen we obtain |fn(z)−gn(z)|<1− |z| n(z∈Fn),(1) where gn:Fn→Cis the function defined as gn(z) =    Pn(z) if z∈B(0,n n+1) qjif z∈Kp(n,j)and p(n, j)≥J(n) 0 if z∈Kp(k,j)(k=n) and p(k, j)≥J(n). We have denoted here by {qj}∞ j=1 any fixed dense sequence in C. Observe that, trivially, gnis continuous on Fnand holomorphic in F0 n, so Nersesjan’s theorem applies properly. Let us define Das the linear span D= span {fn:n∈N}. Of course, Dis a linear submanifold of H(D), and Dis dense because {fn}∞ n=1 is. Indeed, from (1) we have that |fn(z)−Pn(z)|<1 nfor all z∈B(0,n n+ 1). 4 Then if we fix a function P∗ mthere exists a sequence n1< n2<· · · with Pnj=P∗ mfor all j∈N. Now if K⊂Dis compact then there is j0∈Nsuch that K⊂B(0,nj nj+1 ) for every j > j0. Therefore |fnj(z)−P∗ m(z)|<1 nj for all z∈Kand all j > j0, so fnj→P∗ m(j→ ∞) uniformly on compacta in H(D). Hence the closure of {fn: n∈N}in H(D) contains the dense set {P∗ m:m∈N}, which proves the density of {fn}∞ n=1. It remains to show that for every prescribed curve γ∈Gas in the hypothesis and for every function f∈ D\{0}we have Cγ(f) = C. Note that for such function fthere exist N∈Nand complex scalars λ1, . . . , λNsuch that λN= 0 and f=λ1f1+· · · + λNfN. Since Osc (γ)=Tand γshould escape towards T, this curve must intersect all spirals Kjexcept finitely many of them; indeed, if this were not the case then the shape of Kj’s together with the continuity of γwould force γto make infinitely many windings around the origin while approaching T, which would contradict the hypothesis Osc (γ)=T. Therefore there exists j0∈Nsuch that p(k, j0)≥J(N) (k= 1, . . . , N) and γ∩Kp(N,j)=∅(j≥j0). Choose points zj∈γ∩Kp(N,j)(j≥j0). Then by (1) we obtain, for every j≥j0, |fN(zj)−qj|=|fN(zj)−gN(zj)|<1− |zj| N≤1− |zj| ≤ 1−rj and |fn(zj)|=|fn(zj)−gn(zj)|<1− |zj| n≤1−rj(n= 1, . . . , N −1). Hence we get |f(zj)−λNqj|=|λ1f1(zj) + · · · +λNfN(zj)−λNqj| ≤ |λN| · |fN(zj)−qj|+ N−1 ∑ n=1 |λnfn(zj)| <(N ∑ n=1 |λn|)(1 −rj)→0 (j→ ∞). But since λN= 0 the sequence {λNqj:j∈N}is dense in C, so for given α∈C there is a sequence {j1< j2<· · ·} ⊂ Nwith λNqjk→αas k→ ∞. Now we can select a sequence {k(1) < k(2) <· · ·} ⊂ Nand a point t∈Twith wl:= zjk(l)→t (l→ ∞). Then {wl}∞ l=1 ⊂γand f(wl) = f(wl)−λNqjk(l)+λNqjk(l)→α(l→ ∞), so α∈Cγ(f). In other words, Cγ(f) = C, as required. 5 In view of Theorem 2.1, two natural questions arise, namely: (a) Is it possible to replace the arbitrary curve γto an arbitrary sequence {zn}∞ n=1 tending to the boundary (even with Osc ({zn}∞ n=1)=∂G)? The elementary Proposition 2.2 below answers this question in the negative. (b) It is clear that a similar result to Theorem 2.1 falls down if one desires that f belongs to a subspace of bounded functions. But even without this boundedness restriction the statement may be false. For instance, if fis in the Hardy space Hp(see below) of the unit disk then Fatou’s theorem asserts that the radial limit limr→1−f(reiθ) exists and is finite for all θ∈A, where A=Afis a subset of [0,2π] such that the Lebesgue measure of [0,2π]\Ais zero, see [6]. Therefore Cγ(f) is a singleton for each radial curve γ={reiθ :r∈[0,1)}(θ∈A). Nevertheless, making a link to a motivating result mentioned in Section 1, we could ask whether at least for a prescribed countable family of curves in D tending to Tthe assertion of Theorem 2.1 holds in Hp. Theorem 2.5 below will provide this time a positive answer, even without the restriction Osc (γ)=T. Proposition 2.2. If G⊂Cis a bounded domain and f∈H(G)then there are a point t∈∂G, a value A∈Cand a sequence {zn}∞ n=1 ⊂Gtending to tsuch that limn→∞ f(zn) = A. Proof. If fhas infinitely many zeros then the result follows from the Analytic Continuation Principle. Suppose now that fhas finitely many zeros. Define g=f/P, where P≡1 if fhas no zeros whereas P(z)≡(z−a1)· · · (z−ap) if a1, . . . , apare the zeros of f, counting according their multiplicities. Then gis in H(G) and has no zeros. Let us fix a sequence {Kn}∞ n=1 of compact subsets of Gwhich is exhaustive, in the sense that its union is Gand Kn⊂K0 n+1 (n∈N). Without loss of generality, we can suppose K0 1=∅. Choose any point a∈K0 1, so a∈K0 nfor all n. Since ghas no zeros, the Minimum Modulus Principle tells us that the minimum of |g|on Knis attained at some point an∈∂Kn, therefore |g(an)| ≤ |g(a)|. Then |f(an)|=|P(an)| · |g(an)| ≤ M:= |g(a)| · supz∈G|P(z)|(n∈N), where Mis finite because Gis bounded. Summarizing, we have obtained a sequence {an}∞ n=1 ⊂G such that {f(an)}∞ n=1 is bounded. But the exhaustivity property of {Kn}∞ n=1 implies that for a given compact set K⊂Gthere is n0∈Nwith K⊂Kn0, so {an:n>n0} ∩ K=∅, whence the compactness of Gleads us up to a point t∈∂G with bn→t(n→ ∞) for some subsequence {bn}∞ n=1 of {an}∞ n=1. Finally, the boundedness of {f(bn)}∞ n=1 guarantees that f(zn)→A(n→ ∞) for some A∈Cand some subsequence {zn}∞ n=1 of {bn}∞ n=1. 6 Recall that a sequence Tn:X→Y(n∈N) of continuous linear mappings between two topological vector spaces X,Yis called universal or hypercyclic whenever there exists a vector x∈X–called universal for {Tn}∞ n=1– whose orbit {Tnx:n∈N}is dense in Y. By U({Tn}) we will denote the set of such universal vectors. If this set is dense in Ythen we say that {Tn}∞ n=1 is densely universal. See [8] for an excellent survey (updated till 1999) about concepts, history and results related to this topic. The following auxiliary result can be found in [3, Theorem 3.1]. Lemma 2.3. Assume that X,Yare metrizable topological vector spaces and that X is Baire and separable. Suppose that, for each k∈N,T(k) n:X→Y(n∈N)is a sequence of continuous linear mappings between Xand Y. Assume that for every k and every sequence {n1< n2<· · ·} ⊂ Nthe sequence {T(k) nj}∞ j=1 is densely universal. Then there exists a dense linear manifold M⊂Xsuch that M\ {0} ⊂ ∩ k∈N U({T(k) n}). If 0 < p < ∞then the Hardy space Hpis the class of functions f∈H(D) for which ∥f∥p:= sup 0<r<1 (∫2π 0 |f(reiθ)|pdθ 2π)1/p <∞. It becomes a Banach space for 1 ≤p < ∞ when endowed with the norm ∥f∥p. In the nineties P. Bourdon and J.H. Shapiro were able to prove that for p= 2 there is a residual subset of functions f∈Hpfor which the orbit {f◦ψn:n∈N}is dense in Hp, where ψnis the nth-iterate of an automorphism ψof Dwithout fixed points in D(see [4] and [14, Chapter 7], where many results of this kind can be found). Their proof equally works for 1 ≤p < ∞ because it is ultimately based on the facts that except for perhaps one point of T the sequence ψn(z) tends to a constant value α∈Tand that for every β∈ Tthe collection of polynomials vanishing at βis dense in Hp, which in turn is a consequence of Beurling’s approximation theorem, see [6, pages 113–114]. Now we denote by φa (a∈D) the automorphism of Dgiven by φa(z) = z+a 1+az . It is a straighforward exercise to check that if {an}∞ n=1 ⊂Dand an→α∈Tthen φan(t)→α(n→ ∞) for every t∈T\ {−α}. With these hints the interested reader will find no difficulty in proving the following extension of Bourdon-Shapiro’s result. Lemma 2.4. Let be prescribed a number p∈[1,∞)and a sequence {an}∞ n=1 ⊂D tending to a boundary point. Then the functions f∈Hpfor which the orbit {f◦φan: n∈N}is dense in Hpform a residual subset. With the help of the latter two lemmas we can conclude this section by proving the following theorem. We remark that since Hp-convergence is stronger than local uniform convergence, the manifold Dobtained below becomes dense also in H(D). 7 Theorem 2.5. Suppose that p∈[1,∞)and that Γis a countable collection of curves in Dtending to the boundary. Then there is a dense linear manifold Din Hpsuch that Cγ(f) = Cfor every f∈ D \ {0}and every γ∈Γ. Proof. Since Γ is countable, we can write Γ = {γk:k∈N}where each γkis a curve in Dtending to T, whence for every kwe can pick a sequence {a(k) n:n∈N} ⊂ γk tending to some point αk∈T. If {n1< n2<· · ·} ⊂ Nthen we have also that a(k) nj→αkas j→ ∞. Thus by Lemma 2.4 the functions f∈Hpfor which the orbit {f◦φa(k) nj :j∈N}is dense in Hpform a residual (so dense) subset of Hpfor every k∈N. In other words, each sequence {T(k) nj}∞ j=1 (k∈N) is densely universal, where T(k) ndenotes the composition operator f∈Hp7→ f◦φa(k) n∈Hp. But X:= Hp=: Y is a Baire metrizable separable topological vector space, hence Lemma 2.3 yields the existence of a dense linear manifold D ⊂ Hpsuch that D \ {0} ⊂ ∩k∈NU({T(k) n}). Finally, take a function f∈ D \{0}and a curve γ=γk∈Γ. Then f∈ U({T(k) n}), which implies that {f◦φa(k) n:n∈N}is dense in Hp, so in H(D). In particular, the set {(f◦φa(k) n)(0) : n∈N}={f(a(k) n) : n∈N}is dense in {g(0) : g∈H(D)}=C. But {a(k) n:n∈N} ⊂ γand a(k) n→αk∈T, so C⊂Cγ(f) and we are done. 3 Final remarks 1. R. Tenthoff has recently constructed (see [15, Kapitel 3]) a dense set of functions f∈H(D) satisfying the following property: For every t0∈T, every compact subset K⊂Dwith connected complement and every continuous function g: K→Cwith g∈H(K0), there exists a sequence of functions tn:K→ {u t0:u∈[0,1)}–not necessarily holomorphic nor continuous– such that limn→∞ tn(z) = z0for all z∈Kand f◦tn→guniformly on K. If we choose specially K={0}then it is derived the following particular case of Theorem 2.1: There is a dense set of functions f∈H(D) all of whose radial cluster sets Cϱ(f, t0) are maximal. 2. In connection with the last remark the following question arises: Is the set {f∈H(D) : Cϱ(f, t0) = Cfor all t0∈T}residual in H(D)? We do not know the answer, but we are able at least to show the next result: The set {f∈H(D) : Cϱ(f, t0) = Cfor all t0belonging to some residual set A=Af⊂T} is residual in H(D). Indeed, by [1] the functions f∈H(D) with maximal cluster set C(f, t0) at any t0∈Tis residual, and by Collingwood’s maximality theorem 8 (see [5, Theorem 4.8]) if F:D→Cis continuous, γis a curve in Dterminating at 1 (in particular, γcan be the radius [0,1)) and γt:= t·γ(t∈T) then Cγt(F, t) = C(F, t) on a residual set (depending on F) of points ton T. 3. Proposition 2.2 showed that at least for a bounded domain G⊂C, there is no function in H(G) with maximal cluster set along any sequence {zn}∞ n=1 ⊂G tending to the boundary ∂G. However, if we drop the amount of sequences {zn}∞ n=1 then it is possible to get a positive result. Given A⊂C, we denote by A′the set of its accumulation points in C∞. Proposition 3.1. Let Abe a nonrelatively compact subset of a domain G⊂C. Then the set M:= {f∈H(G) : CA(f, t) = Cfor all t∈A′∩∂G} is residual in H(G). Proof. Let {tk}∞ k=1 be a countable dense subset of A′∩∂G. For each k, we choose a sequence {a(k) n}∞ n=1 ⊂Awith a(k) n→tk(n→ ∞). By [2], it is known that the sets {f∈H(G) : C{a(k) n:n∈N}(f, tk) = C}(k∈N) are residual, hence by Baire’s theorem D:= ∩ k∈N {f∈H(G) : C{a(k) n:n∈N}(f, tk) = C} is residual. Let f∈ D and t∈A′∩∂G. If we prove that CA(f, t) = C, then we would have f∈ M. Thus D ⊂ M and Mwould be residual. Let {wn}∞ n=1 be a countable dense subset of C. By induction, we can construct an increasing sequence {mn}∞ n=1 ⊂Nsuch that |f(a(k) mn)−wn|<1 n(k= 1, . . . , n;n∈N).(2) Fix a value w∈C. There is an increasing sequence {in}∞ n=1 ⊂Nwith |win−w|<1 in (n∈N).(3) The point tis an accumulation point of the set {a(k) min:k= 1, . . . , in;n∈N} 9