Universal entire functions for affine endomorphisms of C N

1 0 Universal entire functions for affine endomorphisms of CN L. Bernal-Gonz´alez Abstract In this paper the affine endomorphisms of CNwhich support compositionally universal entire functions are completely characterized. 1 Introduction Throughout this paper Nwill denote the set of all positive integers, and if N∈Nthen CNwill stand for the N-dimensional complex space. In particular, C1is the complex plane C. The closed polydisk of radius r≥0 centered at the origin is D(r) = {z= (z1, . . . , zN)∈CN:kzk ≤ r}, where kzk= max1≤j≤N|zj|. A domain Gof CNis a nonempty connected open subset of CN. By H(G) we denote the space of holomorphic functions f:G→C, endowed with the topology of uniform convergence in compacta. In particular, H(CN) is the space of entire functions of Ncomplex variables. It is well known that H(G) becomes a separable Fr´echet space under the above topology. The symbol Aut (G) will stand for the group of all automorphisms (= biholomorphic bijective selfmappings) on G. In 1929 Birkhoff [8] constructed an entire function which is ‘universal’ for translations. In fact, he proved essentially that given b∈C\{0}there exists ∗Partially supported by Plan Andaluz de Investigaci´on Junta de Andaluc´ıa FQM-127. 2000 Mathematics Subject Classification: Primary 47A16. Secondary 32E30, 47B38. Key words and phrases: universal entire function, composition operator, endomorphism of CN, holomorphic convexity. Author’s address: Departamento de An´alisis Matem´atico. Avda. Reina Mercedes, Apdo. 1160. 41080-Sevilla, Spain. E-mail: lb[email protected]. 1 a function f∈H(C) such that its sequence of translates {f(·+nb) : n∈N} is dense in H(C). Birkhoff’s theorem can be observed under the point of view of the operator theory as a universality result; namely, if ϕ:C→Cdenotes the translation z7→ z+bthen the composition operator Cϕ:f∈H(C)7→ f◦ϕ∈H(C) is universal. In general, if Xis a (necessarily separable) topological vector space and Tis an operator (= continuous linear selfmapping) on Xthen (Tn) is said to be universal (or hypercyclic) provided that there exists some vector x∈X–called universal for T– for which the orbit {Tnx:n∈N} of xunder Tis dense in X. Here (Tn) represents the sequence of iterates T1=T, T2=T◦T, . . . of T. It is easy to see that the set of universal vectors is dense. The operator Tis called hereditarily universal if and only if given a sequence {n1< n2<· · · } ⊂ Nthere is a vector x∈Xsuch that {Tnkx:n∈N}is dense in X. If Xis Baire and metrizable and Tis universal (hereditarily universal) then the set of universal vectors for T(for each sequence (Tnk), respectively) is residual, that is, its complement is of first category. These notions can be easily extended to a sequence (Tn) of operators. See [16] for a good account about these concepts and their history. Since 1929 many papers have dealt with the subject of universality through translations in one (complex) variable, but only a few ones in several variables. Let us make a brief report, now in the language of the universality of operators; see also the survey [16] –specially its Section 4a– which contains a rather complete list of references including domains G6=Cand spaces X6=H(G). In 1976 Luh [21] proved that for a prescribed unbounded sequence (bn)⊂Cthe sequence (Cϕn) is universal on H(C), where ϕnis the translation z7→ z+bn. In 1984 Duyos-Ruis [11] showed by functional analysis methods that Cϕ(ϕ(z) = z+b, b ∈C\ {0}) is universal on H(C) (hence there is a residual subset of universal functions), while the residuality of the (Cϕn)-universal entire functions (where the ϕnare the above translations) was observed by Grosse-Erdmann [15] and Gethner and Shapiro [12]. In 1995 Bernal and Montes [6] were able to show the same result for a sequence {ϕn(z) = anz+bn:n∈N} ⊂ Aut (C); recall that ϕ∈Aut (C) if and only if ϕis a nonconstant affine endomorphism of C, that is, there are a, b ∈Cwith a6= 0 and ϕ(z) = az +b. Specifically, they proved that (Cϕn) is universal 2 if and only if the sequence {min{|bn|,|bn/an|} :n∈N}is unbounded, from which they derived that if ϕis an affine endomorphism of Cthen Cϕis universal if and only if ϕis a translation, that is, ϕ(z)≡z+bfor some b∈C\{0}. As for several variables, the history is not too long. In 1941 Seidel and Walsh [23] proved a non-Euclidean version of Birkhoff’s theorem for the unit disk, and in 1979 Chee [10] extended this to the unit polidisk and ball of CN. Le´on [20] has recently characterized the corresponding universal sequences of automorphisms in both domains. In the case of the euclidean translations ϕ(z) = z+b(z∈CN, b ∈C\ {0}) in several variables –where z= (z1, . . . , zN), b = (b1, . . . , bN), and these will be the standard representations of z, b along the current paper– the natural extensions of the theorems of Birkhoff and Luh are covered by [13, Section 5] and by the recent papers [1], [3, Section 2] and [7] (see also [2] and [3] for a matricial extension of Zappa’s result [24], which in turn is a multiplicative version in C\ {0}of Birkhoff’s theorem). In view of the above discoveries, it is natural to pose the problem of characterizing those mappings ϕ∈Aut (CN) such that Cϕis universal on H(CN). Nevertheless, a complete description of Aut (CN) (see for instance [4], [5] and [22] for a study of some subfamilies of it) is unknown up to date. Although there are plenty of automorphisms of CN, the simplest among them are with no doubt the affine linear mappings (or ‘affine endomorphisms’) from CNinto itself which are invertible. Each affine endomorphism S=S(A, b) is biunivocally determined by a pair (A, b), where A:= [aij]i,j=1,...,N is a matrix with complex entries and bis a fixed vector of CN; so Sis given by S(z) = Az +bfor all z∈CN. Observe that as we make the calculation S(z) = Az +bit is convenient to consider the vectors zand bas ‘column’ vectors. It is clear that S∈Aut(C) if and only if Sis one-to-one if and only if Sis onto if and only if det(A)6= 0. Hence the main aim of this paper is to characterize the universality of the composition operator CS:H(CN)→H(CN) generated by an affine endomorphism S=S(A, b) in terms of the matrix Aand the vector b. This will be accomplished in Section 3 where we prove, among other things, that CSis universal if and only if Sis univalent and has no fixed point. In Section 2 we present a number of statements that will reveal useful for our goal. 3 2 Several auxiliary results This section is devoted to background material on N-dimensional complex approximation that will be needed for the work of Section 3. From now on Gwill represent a domain in CN. Let us denote by H(G, G) the set of all holomorphic selfmappings ϕ= (ϕ1, . . . , ϕn) on G, that is, ϕ(G)⊂Gand each component ϕj:G→C(j= 1, . . . , N) is a holomorphic function. Then if ϕ∈H(G, G) the composition operator Cϕ:H(G)→H(G) is well defined. Of course, Aut (G)⊂H(G, G). Our first lemma prevents us to use non-injective selfmappings to obtain Cϕ-universality. Lemma 2.1. If ϕ∈H(G, G)and Cϕis universal on H(G)then ϕis oneto-one and has no fixed points. Proof. The results contained in this lemma are well known in the one-dimensional context, see for instance [9, pages 3 and 10]. The proof given on page 10 of that monograph for the necessity of univalence works, word for word, in several variables. Indeed, if ϕidentifies two distinct points aand bof G, then so does the n-th component ϕnof ϕ, and so does f◦ϕnfor each nand each f∈H(G). Thus if gis a limit point of the Cϕ-orbit of f, then g(a) = g(b), hence (because some g∈H(G), namely an appropriate coordinate function, takes different values at aand b) no fholomorphic on Gcan include every function in H(G) in the closure of its orbit. Hence Cϕis not universal. Finally, if a∈Gwere a fixed point for ϕand f∈H(G) were Cϕ-universal then by considering the compact set K={a}the closure in CNof the set {f(ϕn(a)) : n∈N}={f(a)}would be dense in C, which is absurd. A set B⊂CNis said to be H(CN)-convex (see [17] or [19]) whenever e B=B, where e B:= {z∈CN:|f(z)| ≤ sup t∈B |f(t)|for all f∈H(CN)}. The next generalization of Runge’s approximation theorem for several complex variables is a special case of a statement that can be found in [17, Theorem 4.3.2 and following note]. Proposition 2.2. Let fbe a holomorphic function in a neighborhood of an H(CN)-convex compact subset Kof CN. Then there is a sequence (fj)⊂ H(CN)such that fj→f(j→ ∞)uniformly on K. 4 In 1965 Kallin [18] proved an important separation lemma in several variables, from which the following proposition –that will be crucial for our approximation problem– is a particular case. The word “convex” means “geometrically convex”, that is, a set B⊂CNis convex whenever z, w ∈B implies λz + (1 −λ)w∈Bfor all λ∈[0,1]. Proposition 2.3. If Kand Lare disjoint convex compact sets in CNthen K∪Lis H(CN)-convex. In connection with the last proposition we point out that it is not known yet whether the disjoint union of 4 closed balls in CNis H(CN)-convex. The following lemma settles the question of which sequences of automorphisms are adequate to generate universality. Following [6], we say that a sequence (ϕn)⊂H(G, G) is run-away if and only if given a compact set K⊂Gthere exists n0=n0(K)∈Nsuch that K∩ϕn0(K) = ∅; and we say that a function ϕ∈H(G, G) is non-recurrent whenever its sequence (ϕn) is run-away. Lemma 2.4. Suppose that ϕis an affine automorphism of CN. Then Cϕis universal on H(CN)if and only if ϕis non-recurrent. Proof. Let us suppose that Cϕis universal and that, by way of contradiction, ϕis not non-recurrent. Then there is a compact set Ksuch that K∩ϕn(K)6= ∅for all n∈N. Choose a sequence (zn)⊂Kwith (ϕn(zn)) ⊂Kand a Cϕ-universal function f∈H(G). Consider the constant function g(z) = 1 + maxK|f|. We have that, for every n∈N, max z∈K|g(z)−f(ϕn(z))| ≥ ||g(z)|−|f(ϕn(zn))|| = 1+max K|f|−|f(ϕn(zn))| ≥ 1, which is a contradiction. Conversely, assume that ϕis non-recurrent. Our final goal is to show that the set Mof universal functions for Cϕis residual (so nonempty). Since H(G) is a second-countable Baire space, Birkhoff’s transitivity theorem (see for instance [14, 9.20]) asserts that Mis a dense Gδ-subset (hence residual) if and only if for every pair of nonempty open subsets Aand Bof H(CN) there exists some m∈Nwith (Cϕ)m(A)∩B=∅.(1) 5 Note that (Cϕ)n=Cϕn(n∈N). Fix A, B as before. Then there exists ε > 0, R > 0 and f, h ∈H(CN) such that A⊃A1:= {g∈H(CN) : maxz∈D(R)|g(z)−f(z)|< ε}and B⊃B1:= {g∈H(CN) : maxz∈D(R)|g(z)− h(z)|< ε}. Since ϕis non-recurrent, there exists m∈Nsuch that D(R)∩ ϕm(D(R)) = ∅. But ϕm(D(R)) is a convex compact set because ϕmis continuous and convex-preserving (that is, ϕm(C) is convex whenever Cis convex; this is true because ϕis affine). Then L:= D(R)∪ϕm(D(R)) is H(CN)-convex by Proposition 2.3. Fix Uand Vopen subsets in CNsuch that D(R)⊂U,ϕm(D(R)) ⊂V(so U∪V⊃L) and U∩V=∅. Define the function F:U∩V→Cas F(z) = f(z) if z∈U h(ϕ−m(z)) if z∈V, which is holomorphic on U∪V. Hence by Proposition 2.2 there exists an entire function gsatisfying |F(z)−g(z)|< ε for all z∈L. But from the definition of Fwe get |g(z)−f(z)|< ε for all z∈D(R). and |g(z)−h(ϕ−m(z))|< ε for all z∈ϕm(D(R)). The last display is clearly equivalent to |g(ϕm(z)) −h(z)|< ε for all z∈D(R). In other words, g∈A1and (Cϕ)mg∈B1. Thus, g∈Aand (Cϕ)mg∈B, so (1) holds. Remark 2.5. The proof of Lemma 2.4 can be easily modified to obtain the following extension: Suppose that Gis a convex domain of CNand that (ϕn) is a sequence in Aut(G) of convex-preserving mappings. Then (Cϕn) is universal if and only if (ϕn) is run-away. Moreover, in this case there exists a residual subset of universal functions. Suffice to say that if Gis convex then the polidisks D(R) of the proof of Lemma 2.4 can be replaced to convex 6 compact subsets of Gand that Birkhoff’s transitivity theorem also works with a sequence (Tn) of mappings from a Baire space into a second-countable space, see [16, Theorem 1]. It might be interesting to investigate whether the last lemma can be extended to non-convex domains or to sequences of automorphisms that do not preserve convexity. 3 Universal functions for endomorphisms From now on Awill represent an (N×N)-matrix with complex entries and bwill be a fixed vector in CN. Lemma 2.4 focuses attention on the dynamics of affine mappings of CN. In order to characterize when they generate universal composition operators, we need to know which of such mappings are non-recurrent. We are now ready to state our main result. Theorem 3.1. Assume that S:CN→CNis an affine endomorphism, say Sz =Az +b(z∈CN). Consider the composition operator CSgenerated by S. Then the following properties are equivalent: (a) Shas no fixed point in CNand det(A)6= 0. (b) The vector bis not in ran(A−I)and det(A)6= 0. (c) CSis universal. (d) CSis hereditarily universal. Proof. (a) ⇐⇒ (b): Simply observe that b∈ran(A−I) if and only if there exists z0∈CNsuch that (A−I)z0=bif and only if Az0+b=z0for some z0∈CNif and only if Sz0=z0for some z0∈CN. (d) =⇒(c): This is trivial. (c) =⇒(a): If CSis universal then Sis one-to-one (hence det(A)6= 0) and has no fixed point by Lemma 2.1. (b) =⇒(d): Since det(A)6= 0, Sis an affine automorphism of CN. Let us prove that Sis non-recurrent. Denote by Jthe Jordan matrix of A. Then there is an invertible (N×N)-matrix Qsuch that A=QJQ−1. Define c:= Q−1band Mz := Jz +c(z∈CN). 7 It is easy to see that b6∈ ran(A−I) if and only if c6∈ ran(J−I). On the other hand, non-recurrence is preserved by similarities; more precisely, if ϕ∈H(CN,CN) and ψ∈Aut(CN), then ϕis non-recurrent if and only ψ◦ϕ◦ψ−1is non-recurrent. Since S=Q◦M◦Q−1we obtain that Sis non-recurrent if and only if Mis. The matrix Jis a direct sum of Jordan blocks Jj. The vector chas components corresponding to each of these blocks and to say c6∈ ran(J−I) is to say that at least one of these component, say cjis not in ran(Jj−Ij). Here Ijis the identity matrix having the same dimension as Jj. Let Mjbe the restriction of Mto the direct summand of CNon which Jjacts, i.e., Mjzj=Jjzj+cj. It suffices to prove that Sjis non-recurrent (this follows from the fact that if CNis represented as a product space, then each compact subset of CNis contained in a product of compact subsets corresponding to the factors of CN). Hence we may drop the subscripts and assume that J itself is a Jordan block with c6∈ ran(J−I). In particualr, J−Iis not invertible. Thus the spectrum of Jis the singleton {1}, so Jitself has just 1’s on the main diagonal and the first superdiagonal, and zeros elsewhere. But, by induction, one obtains Mnz=Jnz+ n−1 X k=0 Jkc(z∈CN, n ∈N). The “1” in the (N, N) position is crucial here; it is also the (N, N) entry of any power of M, and in all these powers the rest of the N-th row consists of zeros. Thus (Mnz)N, the N-th component of Mnz, is zN+ncN. Now c6∈ ran(J−I) means cN6= 0, hence as n→ ∞ the N-th component of Mnzgoes to infinity uniformly on each compact subset of CN. Hence ||Snz|| → +∞(n→ ∞) in the same way. It is easy to see that because of this Mis recurrent. Consequently, Sis non-recurrent and, due to Lemma 2.4, CSis universal. Finally, observe that if we fix a sequence {n1< n2<· · · } ⊂ Nthen the same reasoning above –replacing nto nj– shows that (Mnj) (hence (Snj)) is run-away, which by Remark 2.5 proves that (CSnj) is universal. In other words, CSis hereditarily universal. Remarks 3.2. 1. The proof of Theorem 3.1 reveals that, even in the case that Sis not invertible, we have: Shas no fixed point if and only if b6∈ 8