Compositionally universal entire functions on the plane and the punctured plane

Compositionally universal entire functions on the plane and the punctured plane Luis Bernal-Gonz´alez To Professor Tom´as Dom´ınguez Benavides on his 60th birthday Abstract. Necessary/sufficient conditions for a sequence of automorphisms of the complex plane to generate a sequence of composition operators that is universal on the punctured plane are provided. As a consequence, it is derived that only for translations and rotation-dilations there can be entire functions whose orbits present universality. Boundedness of these functions on unbounded sets as well as frequent universality on the whole plane are also analyzed. Mathematics Subject Classification (2010). Primary 30E10; Secondary 47A16, 47B33. Keywords. Universal entire functions, punctured plane, universal sequence of operators, hypercyclic functions, boundedness, frequent universality. 1. Introduction, notation, known results and aim of this paper In 1929 Birkhoff [9] published his celebrated universality theorem asserting the existence of an entire function fon the complex plane Cwhose translates f(·+n) (n∈N:= {1,2, . . . }) approximate uniformly any prescribed entire function on every compact subset of C. His result can be put into the more general setting of universality, whose starting notions are collected in the next paragraph. Assume that Xand Yare topological spaces and that Tn:X→Y(n≥1) is a sequence of continuous mappings. Then (Tn) is said to be universal whenever there exists a point x0∈X, called universal for (Tn), whose orbit {Tnx0:n≥1} under (Tn) is dense in Y. The set of universal points for (Tn) will be denoted by The author has been partially supported by the Plan Andaluz de Investigaci´on de la Junta de Andaluc´ıa FQM-127, by MCI Grant MTM2009-10696-C02-01 and by MCYT (Acci´on Especial) MTM2007-30904-E. 2 Bernal U((Tn)). It is easy to see that, if Xis a Baire space and Yis second countable, then U((Tn)) is dense if and only if it is residual, and if and only if (Tn) is transitive (i.e. given nonempty open sets U⊂X, V ⊂Y, there is n∈Nsuch that Tn(U)∩ V6=∅). If X=Y, then a single continuous selfmapping T:X→Xis called universal (transitive) if the sequence (Tn) of its iterates is universal (transitive, resp.), and we denote U(T) = {universal points for T}:= U((Tn)). Note that U(T)6=∅if and only if U(T) is dense. If now Xand Yare (Hausdorff) topological vector spaces and {Tn:n∈N} ⊂ L(X, Y ) =: {continuous linear mappings X→Y}then the words universal and hypercyclic are synonymous, although “hypercyclic” is mainly used for single mappings T∈L(X) = {operators on X}:= L(X, X). The reader is referred to the surveys [4], [18] and [20] for history, concepts, results and references about universality and hypercyclicity. For a domain (nonempty connected open subset) G⊂C, we denote by H(G) the vector space of all holomorphic functions G→C. Then H(G) becomes an F- space (that is, a complete metrizable topological vector space) when it is endowed with the topology of convergence on compacta; in addition, H(G) is separable, hence second countable. We will denote by M(G) the family of Mergelyan subsets of G, that is, M(G) = {K⊂C:Kis compact and C\Kis connected}. Recall that the sets U(f, ε, K) := {g∈H(G) : |g(z)−f(z)|< ε ∀z∈K}(f∈H(G), ε > 0, K ⊂Ga compact set) form a basis of open subsets in H(G). For the sake of brevity, we introduce the following two concepts. A subset A ⊂ H(G) will be called M-dense provided that for every ε > 0 and every K∈ M(G), one has A ∩ U(f, ε, K)6=∅. By using Runge’s approximation theorem (see [16]), we have that the polynomials form an M-dense subset of H(G). Of course, denseness matches M-denseness if Gis simply connected. If G, Ω are domains in Cand ϕ∈H(G, Ω) (i.e. ϕ∈H(G) and ϕ(G)⊂Ω), then the composition operator Cϕ:H(Ω) →H(G) is the continuous linear mapping defined as Cϕf=f◦ϕ. If (ϕn)⊂H(G, Ω), then we say that the sequence of composition operators (Cϕn) is M-universal provided that there is f∈H(G) –called M-universal for (Cϕn)– such that {f◦ϕn}n≥1is M-dense in H(G). A corresponding notion of M-hypercyclicity arises when G= Ω and each ϕnis the nth-iterate ϕ[n]=ϕ◦ · · · ◦ ϕ(n-fold) of ϕ. In the terminology of the preceding paragraphs, Birkhoff’s theorem tells us that the translation operator f7→ f(·+1) is hypercyclic on H(C). With essentially the same proof as in [9] we have (see [22]) that if (an)⊂Cis an unbounded sequence then the sequence of translations τan:f∈H(C)7→ f(·+an)∈H(C) (n∈N) is universal (a function f∈H(C) (τan)-universal for some (an)⊂C will be called Birkhoff-universal). In 1988, Zappa [28] replaced the additive group (C,+) by the multiplicative group (C∗:= C\ {0},·) (see [1] and [2] for extensions to complex special or general linear groups) and demonstrated the existence of a “multiplicatively universal” function f∈H(C∗), that is, given g∈H(C∗), ε > 0 and K∈ M(C∗), there exists a∈C∗such that |f(az)−g(z)|< ε for all z∈K (by Mergelyan’s approximation theorem [16, Chap. 3], one may in fact prescribe g∈A(K), the space of continuous functions f:K→Cthat are holomorphic Universal entire functions 3 in the interior K0of K). The same approach of [28] shows that if (an)⊂C∗is an unbounded sequence then (Cz7→anz) is M-universal on H(C∗), that is, there is an f∈H(C∗) whose sequence of dilation-rotations {f(an·)}n≥1is M-dense in H(C∗). Denote by Aut(G) the group of holomorphic automorphisms of a domain G. Recall that Aut(C) = {z7→ az +b:a∈C∗, b ∈C}and Aut(C∗) = {z7→ az :a∈ C∗}∪{z7→ a/z :a∈C∗}. Birkhoff’s and Zappa’s results have been extended or improved in several directions. Restricting ourselves to Cand C∗, we remark the following: (a) In 1995, Montes and the author [7] proved that, for a sequence {zϕn 7→ anz+bn}n≥1⊂Aut(C), the sequence of composition operators Cϕn: f7→ f(an·+bn) (n≥1) is universal on H(C) if and only if the sequence {min{|bn|,|bn/an|}}n≥1(1) is unbounded. If this is the case, then there is a residual set of universal entire functions. In particular, the composition operator f7→ f(a·+b) associated to the automorphism z7→ az +bis hypercyclic if and only if a= 1 and b6= 0. Moreover, given (an)⊂C∗, the sequence of composition operators f∈H(C∗)7→ f(an·)∈H(C∗) is M-universal if and only if the sequence {max{|an|,|1/an|}n≥1 is unbounded, in which case there is a residual subset in H(C∗) of M-universal functions. In particular, the operator f7→ f(a·) is M-hypercyclic on H(C∗) if and only if |a| 6= 1. By contrast, no sequence (ϕn)⊂Aut(C∗) generates a universal sequence (Cϕn) on H(C∗), see [7] and [19]. (b) In 1996, Luh [23] improved Zappa’s theorem by constructing, for a given unbounded sequence (an)⊂C∗, an entire function fsuch that {f(an·)}n≥1 is M-dense in H(C∗). The set of such functions fis in fact residual in H(C) [8]. Notice that the unboundedness of (an) is necessary for the existence of these multiplicatively universal entire functions, for if |an| ≤ M < +∞ (n≥1) then the sequence (f(an·)) cannot approximate the constant function 1 + sup|z|≤M|f(z)|on the compact set {1}. In view of (a)–(b), a natural question arises: which sequences {anz+bn}n≥1⊂ Aut(C) support entire functions fsatisfying that {f(an·+bn)}n≥1is M-dense in H(C∗)? We provide necessary as well as sufficient conditions in Section 2. In the last few years, a number of authors have invested much effort in making universality compatible with some additional property, specially boundedness on large sets. In this respect, Tenthoff [26] proved in 2000 that there are Birkhoffuniversal entire functions that are bounded on every (straight) line. M.C. Calder´on [13] (see also [17]) showed in 2002 the existence of a dense linear manifold of Birkhoff-universal entire functions f(see [10] for similar results in the harmonic setting) which are bounded on any sector {z: 0 ≤arg z≤α}(0 < α < 2π) and any strip (i.e. any domain lying between two parallel lines); even exp(|z|α)f(z)→0 4 Bernal (z→ ∞) on these subsets for fixed α∈(0,1/2), and many more additional properties hold. In 2003, Costakis and Sambarino [14] proved that there are τ1- hypercyclic entire functions ftending to 0 as z→ ∞ on each sector {z:ε≤ arg z≤2π−ε}(0 < ε < π). In 2006, Niess [25] gave the following necessary and sufficient condition for the existence of (τan)-universal entire functions bounded on every line: there exists a subsequence (ank) such that, for every R > 0 and every line L, there is k0∈Nwith L∩B(ank, R) = ∅for all k≥k0(as usual, B(a, R) := {z∈C:|z−a|< r}and B(a, R) := {z∈C:|z−a|< r}). Gharibyan, Luh and Niess [17, Theorem 1.2] established the existence of an entire function ϕand of sequences (an),(bn)⊂Cwith an→0, bn→ ∞ such that ϕis (Canz+bn)-universal and bounded on every line. Bonilla and the author [5] proved that, for given F⊂C, there exist a Birkhoff-universal entire function that is bounded on Fif and only if there exists an Arakelian subset F0of C(i.e. F0is closed and C∞\F0is connected and locally connected at ∞, where C∞is the onepoint compactification of C) such that ρi(C\F0) = +∞(here ρi(A) = sup{r > 0 : there exists a closed ball Bof radius rwith B⊂A}, the inscribed radius of a subset A⊂C). In 2010, Calder´on, Luh and the author [6] stated the following: if A⊂Cis an unbounded Arakelian set with ρi(C\A)=+∞, there is a dense linear manifold Mof entire functions all of whose nonzero members are Birkhoffuniversal and exp(|z|α)f(z)→0 (z→ ∞, z ∈A) for all α < 1/2 and f∈M. Passing to C∗, A. Vogt [27] has recently constructed a multiplicative universal entire function ϕwith respect to a given unbounded sequence (an) such that ϕis bounded on some curve Γ tending to ∞. On the contrary, he has proved that any such ϕis necessarily unbounded on every line. In Section 3, we investigate boundedness and unboundedness on large subsets of Cfor entire functions being compositionally universal (M-universal) on H(C) (on H(C∗), resp.) with respect to sequences in Aut(C). In 2006, Bayart and Grivaux [3] presented the more stringent notion of frequent hypercyclicity, that is a kind of quantified hypercyclicity. If Xis a topological vector space, then an operator T∈L(X) is said to be frequent hypercyclic provided that there is a vector x∈X, called frequent hypercyclic for T, such that, for every nonempty open set U⊂X, the set {n∈N:Tnx∈U}has lower positive density, that is, lim inf n→∞ card {k∈ {1, . . . , n}:Tkx∈U} n>0. (2) As in the mere hypercyclicity, the concept can be extended to a sequence of continuous mappings Tn:X→Y(n∈N) between two topological spaces X, Y (simply replace Tkxby Tkxin (2)), in which case (Tn) is called frequent universal, and xa frequent universal element for (Tn), see [12]. Sufficient conditions for frequent hypercyclicity/universality have been given in [3] and [12]. In [3] it is proved the frequent universality of any translation operator τa(a6= 0) on H(C). This is also shown by Bonilla and Grosse-Erdmann in three different ways (see [11], [12, Example 3.2] and [12, Theorem 4.2]) as a consequence of more general criteria. Universal entire functions 5 Our third and final goal, which will be performed in Section 4, is to extend this result by analyzing the frequent universality of a sequence of composition operators on H(C) generated by a sequence (ϕn)⊂Aut(C). 2. M-universal entire functions Our first theorem extends and unifies the assertions given by Luh, Prado and Bernal about existence of entire functions that are M-universal on H(C∗) with respect to sequences of composition operators generated by automorphisms of C, see Section 1. As usual, we adopt the convention α/0=+∞whenever α∈(0,+∞). Theorem 2.1. Assume that {ϕn}n≥1⊂Aut(C)and ϕn(z) = anz+bn(n≥1). (a) If the sequence {max{min{|an|,|an/bn|},min{|bn|,|bn/an|}}}n≥1(3) is unbounded then the sequence Cϕn:H(C)→H(C∗) (n∈N)is M- universal. In fact, the set UM((Cϕn)) := {f∈H(C) : (f◦ϕn)is M-dense in H(C∗)}is a residual subset of H(C). (b) If the sequence Cϕn:H(C)→H(C∗) (n∈N)is M-universal then (i) at least one of the sequences (an),(bn)is unbounded, and (ii) there exists a sequence {n1< n2<· · · } ⊂ Nsuch that the sequence of zeros {ϕ−1 nk(0)}k≥1is not contained in any set K∈ M(C∗). Proof. (a) Assume that (3) is unbounded. Then at least one of the sequences {min{|an|,|an/bn|}}n≥1,{min{|bn|,|bn/an|}}n≥1is unbounded. If the latter one is unbounded, then by [7, Proposition 2.3 and Theorem 3.1] the set U((Cϕn)) = {f∈ H(C) : the set {f◦ϕn}n≥1is dense in H(C)}is a residual subset of H(C). But by Runge’s theorem H(C) is M-dense in H(C∗). Therefore UM((Cϕn)) ⊃ U((Cϕn)), so UM((Cϕn)) is residual in H(C). Suppose now that {min{|an|,|an/bn|}}n≥1is unbounded. Fix a compact set K⊂C, a number ε > 0, a set L∈ M(C∗) and functions f∈H(C), g ∈H(C∗). Consider the sets U(f, ε, K) := {h∈H(C) : |h(z)−f(z)|< ε for all z∈K}and V(g, ε, L) := {h∈H(C∗) : |h(z)−g(z)|< ε for all z∈L}, which are respectively basic open sets in H(C) and H(C∗). Choose r, s > 0 with K⊂B(0, r) and L∩B(0,2s) = ∅, then select n∈Nsuch that |an|> r/s and |an/bn|>1/s. Then we obtain, for all z∈L, |ϕn(z)|=|anz+bn|=|an|z+bn an>r s(2s−s) = r. Hence B(0, r)∩ϕn(L) = ∅. Define the function F:B(0, r)∪ϕn(L)→Cby F(z) = f(z) if z∈B(0, r) g(z−bn an) if z∈ϕn(L). Note that S:= B(0, r)∪ϕn(L)∈ M(C) because ϕn∈Aut(C). Moreover, Fis holomorphic on a neighborhood of S. By Runge’s theorem, there exists a polynomial h (so h∈H(C)) with |h(z)−F(z)|< ε for all z∈S. In particular, |h(z)−f(z)|< ε 6 Bernal for all z∈Kand |h(ϕn(z))−g(z)|< ε for all z∈L. In other words, h∈U(f, ε, K) and Cϕnh∈V(g, ε, L). Therefore Cϕn(U(f, ε, K))∩V(g, ε, L)6=∅, whence each set Sn∈NC−1 ϕn(V(g, ε, L)) is (open and) dense in H(C). Now, for every domain Gthere exists a sequence {Kj}j≥1⊂ M(G) that is exhaustive, that is, given L∈ M(G) there is j0∈Nwith L⊂Kj0[7, Lemma 2.9]. For the special case G=C∗, it follows that UM((Cϕn)) = \ g∈H(C∗) L∈M,ε>0[ n∈N C−1 ϕn(V(g, ε, L)) = \ k,l,j∈N[ n∈N C−1 ϕn(V(gk,1/l, Kj)), where (gk) is any fixed dense sequence in H(C∗). Since H(C) is a Baire space, we obtain that UM((Cϕn)) is a dense Gδ(so residual) subset of H(C). (b) Suppose that there exists f∈H(C) that is M-universal for the sequence Cϕn:H(C)→H(C∗) (n∈N). By way of contradiction, assume also that (an) and (bn) are bounded. Then |an|+|bn| ≤ M < +∞(n∈N), say. Therefore no sequence (f(ankz+bnk)) would be able to approximate the constant function 1 + max|w|≤M|f(w)|on the compact set {1} ∈ M(C∗), so contradicting the hypothesis. Thus, (i) is satisfied. Suppose now, again by way of contradiction, that (ii) does not hold. By hypothesis, there is f∈H(C) as well as a sequence {n1< n2<· · · } ⊂ Nsuch that f(ankz+bnk)→g(z) (k→ ∞) uniformly on each member of M(C∗), where g(z) := 1 + |f(0)|. But one can find a set K∈ M(C∗) with {ϕ−1 nk(0) = −bnk/ank}n≥1⊂K. Then sup z∈K |f(ankz+bnk)−g(z)|≥|f(ϕnk(ϕ−1 nk(0))) −g(z)|=|f(0) −(1 + |f(0)|)| ≥ 1 for all k∈N, which is absurd. This proves the theorem.  Corollary 2.2. Let ϕ∈Aut(C), with ϕ(z) = az +b. Then the following properties are equivalent: (a) There is an entire function fwhose orbit {f◦ϕ[n]}n≥1is M-dense in H(C∗). (b) The set UM((Cϕ[n])) is residual in H(C). (c) Either |a|>1and b= 0 or a= 1 and b6= 0. Proof. It is evident that (b) implies (a). Suppose that (c) holds. If a= 1 and b6= 0 (i.e. ϕis a translation) then we have in fact that Cϕis hypercyclic on H(C), so the sequence (Cn ϕ)=(Cϕ[n]) is transitive from H(C) into itself. Therefore the set U((Cn ϕ)) of its hypercyclic functions is residual in H(C). But, again by Runge’s theorem, U((Cn ϕ)) ⊂ UM((Cϕ[n])), from which the residuality of UM((Cϕ[n])) follows. If a6= 1 then ϕ[n](z) = anz+an−1b+an−2b+· · · +ab =anz+bn, where an=anand bn=b·an−1 a−1. Therefore, if |a|>1 and b= 0, the sequence {min{|an|,|an/bn|}n≥1={|a|n}n≥1is unbounded. By Theorem 2.1, (b) is satisfied. Finally, if (a) is true and a6= 1 then by Theorem 2.1 at least one of the sequences (an),(ban−1 a−1) is unbounded, hence |a|>1. If a6= 1 and b6= 0 then (ϕ[n])−1(0) = b(1−a−n) 1−a−→ n b 1−a∈C∗, so the full sequence {(ϕ[n])−1(0)}n≥1is contained in the set K:= {b 1−a}∪{(ϕ[n])−1(0) : n∈N} ∈ M(C∗). According to Universal entire functions 7 Theorem 2.1, there is no entire function fsuch that {f◦ϕ[n]}n≥1is M-dense. If a= 1 and b= 0 then ϕis the identity, whose iterates clearly do not generate compositional universality. Hence (a) implies (c).  Remark 2.3.1. As an example, there is no entire function fsuch that the set A(f) := {z7→ f(n2z+n) : n∈N}is dense in H(C), because the sequence (1) (see Section 1) for an=n2, bn=nis {1/n}n∈N, which is bounded. Nevertheless, entire functions fsuch that A(f) is M-dense in H(C∗) do exist: indeed, min{|an|,|an/bn|} =n(n∈N) and Theorem 2.1 applies. 2. Let (qn) be an enumeration of the set {eit :t∈Q}(Q= the set of rational numbers), which is dense in the unit circle T. Then ϕn(z) = anz+bn:= nz +nqn∈Aut(C) (n≥1) satisfies that the sequences (1) and (3) are bounded, (an) and (bn) are unbounded and the full sequence (ϕ−1 n(0)) = (−qn) is not contained in any member of M(C∗). Hence (Cϕn) is not H(C)-universal but we can extract from Theorem 2.1 neither the M-universality nor the non-M-universality of (Cϕn) on H(C∗). This raises the question of finding an intermediate property between the unboundedness of (3) and (i)+(ii) of Theorem 2.1 characterizing the M-universality of {H(C)Cϕn −→ H(C∗)}n≥1. 3. Bounded universal entire functions In this section we show that, under appropriate conditions, no M-universal entire function on H(C∗) is bounded on every member of a large family of unbounded sets (including lines), see Theorem 3.1. In the positive direction, it will be proved (Theorem 3.3) the existence of an M-universal entire function ffor which there is an unbounded set Γ such that ftends to 0 as z→ ∞ (in particular, fis bounded) on it. We may even prescribe the set Γ within a large family, including again all lines, in the case of H(C)-universality, see Theorem 3.5. This improves recent results by A. Vogt [27] and completes a number of findings by several authors, see Section 1. We introduce the following notion. By S0we denote the collection of all closed sectors Swith vertex at the origin and amplitude ampl(S)∈(0,2π). We say that a set Γ is totally non-spiral-like provided that there are two unbounded connected sets Γ0, L and a sector S∈ S0such that Γ0⊂Γ, L⊂Sand Γ0∩L=∅. For instance, every set Γ with non-total oscillation near ∞(i.e. such that there are R > 0, S∈ S0and an unbounded connected set Γ0⊂Γ with Γ0∩S∩{|z|> R}=∅) is totally non-spiral-like. In particular, every line, every parabola, and in general every unbounded algebraic curve P(x, y) = 0 (P= a nonconstant polynomial of two real variables with real coefficients) are totally non-spiral-like sets. Theorem 3.1. Assume that {ϕn(z) = anz+bn}n≥1⊂Aut(C)is a sequence such that an→ ∞ and bn/an→0as n→ ∞. Let Γbe a totally non-spiral-like subset of C. Then every M-universal entire function for the sequence Cϕn:H(C)→H(C∗) (n≥1) is unbounded on Γ. 8 Bernal Proof. Consider the compact set given by the “snail” curve K:= {z(θ) = 1 + θ 2πeiθ ∈C:θ∈[0,6π]}. Observe that K∈ M(C∗) and note that any ray σstarting from any point in D intersects Kat least at two different points z(θ1,σ), z(θ2,σ) with θ2,σ −θ1,σ > π and |z(θ1,σ)|>1,|z(θ2,σ)|>2. It follows that if Mis an unbounded, connected set lying in a non-degenerated closed sector Σ whose vertex is at Dwith M∩D6= ∅, then Mmust intersect both curvilinear arcs of the closed annular “sector” determined by Kand the segments joining the pair of points (z(θ1,σ), z(θ2,σ)) and (z(θ1,λ), z(θ2,λ)), where σand λare the rays forming the boundary of Σ (there may be more than one of such annular sectors; then simply select one of them). As a consequence, if Ais an unbounded connected subset of Csuch that A∩D6=∅=A∩M, then Ashould intersect the snail, i.e. A∩K6=∅. This property will be used later. Notice that by Theorem 2.1, there are in fact M-universal entire functions for (Cϕn). Let fbe one of them and assume, by way of contradiction, that f is bounded on Γ, say |f(z)| ≤ α(z∈Γ). By universality, there is a sequence {n1< n2<· · · } ⊂ Nsuch that Cϕnkftends to the constant function 1 + α uniformly on Kas k→ ∞. Without loss of generality, we may assume that (nk) is the full sequence N. In particular, there is n0∈Nsuch that sup z∈K |f(anz+bn)−(1 + α)|<1 for all n≥n0.(4) Let Γ0, L, S the sets supplied by the definition of non-total spiral-likeness for our set Γ. Fix any z0∈Γ0and any w0∈L. Since an→ ∞ and bn/an→0, there is m∈Nwith m≥n0such that |z0/am|<1/2,|w0/am|<1/2 and |bm/am|<1/2. Then z0−bm am,w0−bm am∈D. We have: •The sets A:= {z−bm am:z∈Γ0}and M:= {z−bm am:z∈L}are unbounded connected sets, and Σ := {z−bm am:z∈S}is a closed sector with ampl(Σ) ∈(0,2π) and vertex at the point −bm/am∈D. •A∩D6=∅ 6=M∩D, A ∩M=∅and M⊂Σ. From the first paragraph, A∩K6=∅. Therefore there exist z1∈Kand z2∈Γ0 (so z2∈Γ) such that z1=z2−bm am. It follows from (4) that |f(amz1+bm)−(1+α)|< 1. Hence |f(z2)−(1+α)|<1, so |f(z2)|> α due to the triangular inequality. This contradicts the fact that |f| ≤ αon Γ.  Remark 3.2.If the snail Kof the last proof had only two turns (i.e. θ∈[0,4π]) then not every ray σwith vertex at a point of Dwould intersect at least twice the set K: take for instance σ={z=t+i(−1+(t/2)) : t≥0}. In the next result, notice that similarly to Theorem 3.1 the condition imposed on (an),(bn) guarantees M-universality on H(C∗), but discards H(C)-universality. By contrast, the restriction imposed in Theorem 3.4 entails H(C)-universality for the sequence (Cϕn), with ϕn(z) = anz+bn. Universal entire functions 9 Theorem 3.3. Let {ϕn(z) = anz+bn}n≥1⊂Aut(C). Suppose that the sequence {min{|an|,|an/bn|}}n≥1is unbounded. Let ω:C→(0,+∞)be a continuous function. Then there exists an M-universal entire function fwith respect to {Cϕn: H(C)→H(C∗)}n≥1that admits an unbounded connected set Γ⊂Csuch that lim z→∞ z∈Γ ω(z)f(z)=0. Proof. We will follow the approach of [27] in the first steps. Choose an exhaustive sequence {Kn}n≥1in M(C∗) as in the proof of Theorem 2.1 (see [7, Lemma 2.9] or [23, Lemma 3]). Let {Qn}n≥1be an enumeration of all polynomials whose coefficients are in Q+iQ. In addition, let {(Q∗ n, K∗ n)}n≥1be a sequence with the property that every pair (Qi, Kj) (i, j ∈N) occurs infinitely many often. By hypothesis, and without loss of generality, one can assume that min{|an|, |an/bn|} → +∞as n→ ∞. Then an→ ∞ and bn/an→0. Therefore, given r, s ∈(0,+∞), one has |an|(r− |bn/an|)> s for nlarge enough. (5) Set n1= 1. Proceeding by induction, and assuming that n1< n2<· · · < nkhave been already determined, we can choose due to (5) an integer nk+1 > nksuch that |ank+1 |(inf{|z|:z∈K∗ k}−|bnk+1 /ank+1 |)>1 + sup{|z|:z∈ankK∗ k+bnk}.(6) With the help of the triangular inequality, (6) shows that the sets Ak:= ankK∗ k+ bnk(k∈N) are mutually disjoint. Of course, each Akis compact and has connected complement. In addition, due to (6), one gets dist(Aj, Ak)≥1 (j, k ∈N;j6=k). Then Sk∈NAkis closed and G:= C\Sk∈NAkis an unbounded connected open subset of C. Hence we can find an unbounded regular curve Γ in G, that is, Γ is the image of some differentiable function γ: [0,+∞)→Gwith limt→∞ γ(t) = ∞. In particular, Γ is an unbounded connected closed subset of Cwith empty interior Γ0. Consider the set A:= G∪Sk∈NAk. Then Ais a Carleman set, that is, it satisfies the following properties: •Ais closed. •C∞\Ais connected and locally connected at ∞. •For every compact set K⊂Cthere is a neighborhood Vof ∞in C∞such that no component of A0intersects both Kand V. Define the function g:A→Cby g(z) = (0 if z∈Γ Q∗ kz−bnk ankif z∈Ak(k∈N). This function is evidently holomorphic in A0and continuous on A. Moreover, the function ε(z) = 1 (1+|z|)ω(z)if z∈Γ 1 kif z∈Ak(k∈N)is continuous on Aand positive. 16 Bernal [25] M. Niess, Universelle Funktionen mit zus¨atzlichen Eigenschaften. Dissertation an der Universit¨at Trier, 2006. [26] R. Tenthoff, Universelle holomorphe Funktionen mit vorgegebene Approximationswegen. Shaker Verlag, Aachen, 2000. [27] A. Vogt, On bounded universal functions. Preprint, 2011. [28] P. Zappa, On universal holomorphic functions. Bollettino U. M. I. (7) 2-A (1988), 345–352. Luis Bernal-Gonz´alez Departamento de An´alisis Matem´atico Facultad de Matem´aticas, Apdo. 1160 Avda. Reina Mercedes, 41080 Sevilla, Spain e-mail: [email protected]