Backward Φ-shifts and universality

TITLE: BACKWARD Φ-SHIFTS AND UNIVERSALITY. AUTHOR: LUIS BERNAL-GONZ´ ALEZ. AFFILIATION: DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO. FACULTAD DE MATEM´ ATICAS. AVENIDA REINA MERCEDES. APARTADO 1160. 41080 SEVILLA, SPAIN. E-MAIL: lb[email protected]. ABBREVIATED TITLE: BACKWARD Φ-SHIFTS. ADDRESS FOR MANUSCRIPT CORRESPONDENCE: LUIS BERNALGONZ´ ALEZ. DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO. FACULTAD DE MATEM´ ATICAS. AVENIDA REINA MERCEDES. APARTADO 1160. 41080 SEVILLA, SPAIN. E-MAIL: lb[email protected]. FOOTNOTES TO THE TITLE: 1This work has been partially supported by Plan Andaluz de Investigaci´on de la Junta de Andaluc´ıa. 22000 Mathematics Subject Classification: Primary 47B37. Secondary 46A45, 47A16, 47H30, 54H20. 1 Backward Φ-shifts and universality by Luis Bernal-Gonz´alez Abstract In this paper we consider spaces of sequences which are valued in a topological space Eand study generalized backward shifts associated to certain selfmappings of E. We characterize their universality in terms of dynamical properties of the underlying selfmappings. Applications to hypercyclicity theory are given. In particular, Rolewicz’s theorem on hypercyclicity of scalar multiples of the classical backward shift is extended. Key words and phrases: sequence space, universal mapping, backward Φshift, Φ-product map, lpspaces, hypercyclic operator, Rolewicz’s theorem. 1 Introduction In 1969 Rolewicz [26] was able to prove that for any scalar cwith |c|>1 (and only for these scalars) the multiple cB of the backward shift Bon the sequence spaces lp(1 ≤p < ∞) and on c0is universal, that is, there exists some vector with dense orbit. The operator Bis defined as B(x1, x2, x3, . . .) = (x2, x3, x4, . . .). Classical and weighted backward shift operators have been extensively studied during the last two decades in connection with hypercyclicity and chaos, see for instance [12], [22], [27], [3], [7], [14], [19], [15], [20], [18], [21], and the references contained in them. Interest in shift operators comes, among other reasons, from the fact that many classical operators can be regarded as such operators. For instance, the differentiation operator Df =f0on the space H(C) of entire functions 2 on the complex plane Cmay be viewed as the weighted backward shift D: (a0, a1, a2, . . .)7→ (a1,2a2,3a3, . . .). as soon as H(C) is considered as the space of complex sequences (a0, a1, . . .) with |an|1/n →0 (n→ ∞). Here the sequence of weights is 1,2,3, . . .. In [27], [15], [18] and [21], among others, the universality of weighted backward shift operators acting on certain sequence spaces has been completely characterized. In particular, Grosse-Erdmann considers rather general sequence spaces in [15]. A. Peris [25] studied universality and chaos (in the sense of Devaney [9]; in usual settings, chaos is equivalent to universality plus existence of a dense set of periodic points, see [2]) of polynomials P:lq→lqgiven by P(x1, x2, . . .) = (p(x2), p(x3), . . .), where p:C→Cis a complex polynomial. This study was extended to K¨othe sequence spaces in [20, Cap´ıtulo 4] for the cases p(z) = zm and p(z)=(z+ 1)m−1. In this paper we are concerned with the dynamics of a class of (unweighted, this time) backward shift operators, namely, the Φ-shifts, which contains the previously cited cases. A Φ-shift is a map Bf:S→Sgiven by Bf(x1, x2, . . .) := (f(x2), f(x3), . . .), where Sis a certain subspace of EN,Eis a topological space, and f:E→E is a continuous selfmap, see Section 3. We also consider the related notion of Φ-product map Πf. Both kinds of maps will be completely characterized on lp spaces. Our main goal is to characterize the “wild behavior” of a Φ-shift Bfin terms of the dynamics of f. This will be done in Section 4. Finally, we provide in Section 5 some applications to the theory of hypercyclic operators. 2 Universality, discrete dynamical systems and sequence spaces The current section is devoted to fix some notation and to collect some definitions and known results coming from Topological Dynamics and from elementary 3 theory of spaces of sequences. We refer the interested reader to the excellent surveys [14], [16] and [8] for a summary of concepts, history and statements dealing with universality and hypercyclicity. Assume that Xand Yare topological spaces and that Tn:X→Y(n∈N:= {1,2,3, . . .}) is a sequence of continuous mappings. Then (Tn) is said to be universal whenever there exists some element x∈Xwhose orbit {Tnx:n∈N} under (Tn) is dense in Y. In this case xis called a universal element for (Tn). Observe that the universality of (Tn) forces Yto be separable. The sequence (Tn) is called densely universal when the set U((Tn)) of universal elements for (Tn) is dense in X. Finally, (Tn) is said to be topologically transitive (in the sense of Birkhoff) provided that to every pair of nonempty open subsets Uof Xand V of Ythere exists some n∈Nwith Tn(U)∩V6=∅. Assume now that X=Yand that T:X→Xis a continuous selfmapping. From the point of view of the behaviour of the sequence (T◦n) of its iterates –that is T◦1=T, T◦2=T◦T, . . .–Tcan be considered as a “discrete dynamical system”. Then Tis called universal whenever the sequence (T◦n) is universal; in this case the set U(T) := U((T◦n)) of universal elements for Tis dense in X; indeed, if x0is universal for Tthen T(and so each T◦m) has dense range, hence each point T◦mx0is universal. A continuous selfmapping Tis said to be topologically transitive whenever (T◦n) is topologically transitive. Finally, Tis called weakly mixing provided that the mapping T×T: (x, y)∈X×X7→ (Tx, Ty)∈X×X is topologically transitive, where X×Xis assumed to carry the product topology. Furstenberg [11, Proposition II.3] proved that in this case the J-product map T×T× · · · × T(Jtimes): (x1, . . . , xJ)∈XJ7→ (Tx1, . . . , TxJ)∈XJis also transitive for every J. And Banks [1, Lemma 5] has shown that Tis weakly mixing if for given nonempty open subsets U1, U2, V ⊂Xthere is an N∈Nsuch that T◦N(V)∩Uj6=∅for j= 1,2. Therefore we have that Tis weakly mixing if and only if for given finitely many nonempty open subsets U1, . . . , UJ, V ⊂X there is an N∈Nsuch that T◦N(V)∩Uj6=∅for j= 1, . . . , J. This motivates the following definition, which will be used in Section 4. 4 Definition 2.1. Assume that fn:E→E(n∈N) is a sequence of continuous selfmappings on a topological space Eand that ais a point in E. We say that (fn) is weakly mixing at a if and only if for given finitely many nonempty open sets U1, . . . , Um, V such that a∈Vthere exists N∈Nsatisfying fN(V)∩Uj6=∅for all j= 1, . . . , m. And we say that a continuous selfmapping f:E→Eis weakly mixing at a whenever its sequence (f◦n) of iterates is weakly mixing at a. The following universality criterion will reveal useful in Section 4. It can be found in, for instance, [14, Section 1a] (see also [13, Kapitel 1]). Theorem 2.1. Suppose that X, Y are topological spaces, in such a way that X is a Baire space and Yis second-countable. Let (Tn)be a sequence of continuous mappings from Xto Y. Then the following assertions are equivalent: (a) The sequence (Tn)is densely universal. (b) The sequence (Tn)is topologically transitive. In a linear setting, that is, when X, Y are topological vector spaces on K(:= C or the real line R) and Tn(n∈N) (or T) are linear and continuous, the words universal and hypercyclic are synonymous. By an operator we mean a continuous linear selfmapping on a topological vector space. By ωwe denote, as usual, the space of all scalar sequences ω=KN. It becomes a Fr´echet space (= complete metrizable locally convex space) when it is endowed with the metric d(x, y) = ∞ X j=1 1 2j |xj−yj| 1 + |xj−yj|, where x= (xj) and y= (yj). For 0 <p<∞we consider the lpspaces lp={x= (xj)∈ω: ∞ X j=1 |xj|p<∞}. If µ(p) = p(p≥1), µ(p) = 1 (p < 1) and kxkp:= (P∞ j=1 |xj|p)1/µ(p)then for p≥1 the space lpbecomes a Banach space under the norm k · kp, while for p < 1 5 the space lpis an F-space (= complete metrizable linear space) under the metric d(x, y) = kx−ykp. Recall also that the space c0={x= (xj)∈ω: limj→∞ xj= 0}is a Banach space when it is endowed with the norm kxk0:= supj∈N|xj|. Generalizations of this kind of sequences spaces will be considered in Section 3. 3Φ-product maps and backward Φ-shifts In this section and in the next one Ewill denote a Hausdorff topological space, and Swill stand for a subset of the space ENof E-valued sequences x= (xj), so xj∈Efor all j∈N. For every a∈Ewe denote by σ(a) the set of sequences ending with a, that is, σ(a) = ∪∞ J=1σJ(a), where σJ(a) = {x= (xj)∈EN:xj=a for all j > J}. From now on, we will assume that Sis a standard sequence space in the sense established by the next new concept. Definition 3.1. We define a standard sequence space (SSS) on Eas a subset S⊂ENendowed with a topology such that there exists a point a∈Esatisfying the following four properties: (S1) The space Sis Baire and second-countable. (S2) The topology on Sis stronger than that inherited from the product topology on EN. (S3) The set σ(a) is a dense subset of S. (S4) For each J∈N, the topology of each σJ(a) inherited from Sis the product topology. If Sis a SSS and a∈Eis a point satisfying (S3)–(S4) then we will say ais a distinguished point for S. If Eis a Hausdorff topological vector space, then a topological vector space S⊂ENis called a linear standard sequence space on E whenever it satisfies (S1)–(S4) for the point a= 0. Sometimes we will also consider the following property: 6 (S4*) Given J∈N, an open set U⊂Sand a point α∈σJ(a)∩U, there exist open sets U1, . . . , UJ, A in Esuch that α∈Π∞ j=1A(N) j⊂Ufor all N > J, where A(N) j=     Uj(1 ≤j≤J) A(N+ 1 ≤j≤N+J) {a}(J < j ≤Nor j > N +J). (1) Remark 3.1. Due to the presence of U1, . . . , UJ, (S4*) implies (S4). In (S4*), the existence of “sliding J-wagon trains” A×· · ·×Ain the projections of every neighborhood of each point of σJ(a) reveals some “indifference” among the coordinates of the elements of Swhen they are close to a. On the other hand, (S4) implies that for every j∈Nthe immersion ij:t∈E7→ (a, a, . . . , a, t, a, a, . . .)∈S (where toccurs at the jth place) is continuous. Of course, (S2) tells us that convergence in Simplies coordinatewise convergence. Examples 3.2. 1. The spaces ω,lp(0 < p < ∞) and c0are linear SSSs: suffice it to take E=Kwith the usual topology. Property (S1) for these spaces is easily checked just by taking into account that a completely metrizable separable space is Baire and second-countable. The remaining conditions are straightforward. A different example is the space S={x= (xj)∈KN: limj→∞ x2j−1= 0 and P∞ j=1 |x2j|<∞}, which becomes a Banach space under the norm kxk= supj≥1|x2j−1|+P∞ j=1 |x2j|. All these spaces also satisfy (S4*). 2. The direct sum S=⊕n∈NKof countably many lines endowed with the inductive limit locally convex topology is a second-countable topological vector space satisfying (S2) to (S4) for E=Kand a= 0, but it is not a linear SSS since Sis not a Baire space. 3. Let S={x= (xj)∈RN: limj→∞ jxj= 0}. Then Sbecomes a separable Banach space when it is endowed with the norm kxk= supj∈N|jxj|. Then S satisfies (S1) to (S4) (for a= 0; no other point a∈Ris possible), so it is a linear SSS. But (S4*) fails because given a ball U={kxk< ε}then we have diam (πj(U)) ≤2ε/j →0 (j→ ∞), see Remark 3.1. Here, as usual, πjdenotes the j-projection πj:x= (xn)∈S7→ xj∈E(j∈N). 7 Another family of examples of linear SSSs which are extensions of ω, c0, lpis described as follows. Assume that Eis a separable Banach space over Ror C with norm k·k. Consider the spaces of E-valued sequences ω(E), c0(E), lp(E) (0 <p<∞). They are defined as the former spaces just by replacing Kwith E, and the absolute value with k·k. As a matter of fact, in the case ω(E) it is enough to assume that Eis a completely metrizable separable topological space. Now we are going to motivate the new concepts provided in Definition 3.2, see below. As seen in Section 1, Rolewicz [26] proved the universality of the backward shift cB : (xj)7→ (cxj+1) (|c|>1) on c0and lp(1 ≤p < ∞), while GrosseErdmann [15, Corollary 2] noted that any weighted backward shift (in particular, Bitself) is universal, even chaotic, on ω. Mart´ınez and Peris [21] have recently studied backward shifts on K¨othe echelon spaces c0(A), λp(A) (1 ≤p < ∞) (see [17] and [23] for definitions and properties; they are separable Fr´echet spaces and include c0, lp(1 ≤p < ∞) for adequate matrices A), and in particular they characterize the universality of Bin terms of the matrix A[21, Proposition 3.1]. On the other hand, Bernardes [6] showed that for given m > 1 there is no universal m-homogeneous continuous polynomial on any Banach space; in particular, the shift (xj)7→ (xm j+1) is not universal on lp(1 ≤p < ∞) or c0. However, the last mapping is universal, even chaotic, on the Fr´echet space ω=CN[24] and in fact on some (non-Banach, of course) K¨othe spaces λp(A) [20]. Furthermore, Peris [25] proves that the (non-homogeneous) polynomial (zj)7→ ((zj+ 1)m−1) is universal (and chaotic) on the complex Banach spaces lp(1 ≤p < ∞) and c0. This is again true on certain spaces λp(A), see [20]. 8 Definition 3.2. Suppose that Sis a SSS and that T:S→Sis a continuous selfmapping on S. (a) We say that Tis a Φ-product map on Sif there exists a selfmapping f:E→E such that T= Πfon S, where Πfx= (f(xj)) for every x= (xj)∈EN. (b) We say that Tis a backward Φ-shift on Sif there exists a selfmapping f: E→Esuch that T=Bfon S, where Bfx= (f(xj+1)) for every x= (xj)∈EN. Remarks 3.3. 1. Observe that Bf= Πf◦B=B◦Πf, where Bis the ordinary backward shift, i.e. Bx = (xj+1) for x= (xj). Of course, if gis the identity on Ethen Πg= the identity on ENand Bg=B. Note also that if either Πfis a Φ-product map or Bfis a backward Φ-shift on Sthen fis continuous. Indeed, f=π1◦Πf◦i1=π1◦Bf◦i2, where i1, i2are the 1and 2-immersions (see Remark 3.1) –which are well-defined by (S3) and continuous due to (S4)– and π1 is the 1-projection, which is continuous by (S2). 2. Note that even in the case of a linear SSS Son Ethe mapping f:E→E may be nonlinear, so both Πfand Bfmay well be nonlinear. It is interesting to obtain necessary and sufficient conditions for Πf(Bf, resp.) to be well-defined on a SSS S(that is, for Sto be Πf-invariant: Πf(S)⊂S, or respectively, Bf-invariant: Bf(S)⊂S) and to be a Φ-product map or a Φshift (that is, continuous). This will be carried out at least for the most usual spaces ω(E), lp(E) (0 <p<∞), c0(E). Recall that the continuity of fis a general necessary condition for the continuity of Bf. In connection with this, we remark that Wildenberg [28] discovered in 1988 the absence of nontrivial functions f:R→Rfor which a sequence (xj) in RNis summable only if (f(xj)) (= Πf(xj), in our terminology) is summable. Specifically, he stated that the last property holds for any sequence (xj) if and only if there exists a constant kwith f(t) = kt in a neighborhood of the origin. Of course, this is not necessary for Πf(l1)⊂l1: take, for instance, f(t) = t2. In fact, we will use an approach similar to [28, Lemma 1] in order to obtain the Πf-invariance of the spaces lp(E). Lemma 3.4. Let Xbe a topological space. Assume that ais a point in Xand that ϕ:X→[0,∞)is a function such that 9 (c) If (Πfn)is a universal sequence of Φ-product maps on Sthen the sequence (fn)is universal on E. In particular, if Πf:S→Sis a universal Φ-product map then fis universal on E. Proof. (a) Assume that Bfis universal on S, and fix a point y∈E. From (S3), the point y= (y, a, a, a, . . .) is in Sfor some a∈E. By universality, there is a point (xj)∈Sand a sequence (nk) of positive integers such that the sequence ((f◦nk(xj+nk))j∈N) converges to (y, a, a, . . .) in Sas k→ ∞. From (S2), f◦nk(x1+nk)→y, hence f(f◦nk−1(x1+nk)) →y, which proves that yis in the closure of f(E). But ywas arbitrary, so f(E) is dense. (b) From Theorem 3.5, Bfis in fact a backward Φ-shift on ω(E). Recall that ω(E) is a second-countable Baire space. In order to apply Theorem 2.1, take X:= ω(E) =: Y,Tn:= Bn f. Let us try to check the Birkhoff transitivity property. Fix nonempty open subsets U, V of ω(E). Then there exist J∈Nand nonempty open subsets U1, . . . , UJ, V1, . . . , VJin Esuch that U1× · · · × UJ×E×E× · · · ⊂ Uand V1× · · · × VJ×E×E× · · · ⊂ V. Since fhas dense range and is continuous, we have that f◦Jhas also dense range. From this, we derive the existence of points t1, . . . , tJ∈Esuch that f◦J(tj)∈Vj (j= 1, . . . , J). Choose any points yj∈Uj(j= 1, . . . , J) and any point t∈E. Consider the sequence x= (xj)∈ω(E) defined as xj=       yj(1 ≤j≤J) tj−J(J < j ≤2J) t(j > 2J). It is clear that x∈U. Finally, TJx=BJ fx= (f◦J(xj+J)), but f◦J(xj+J) = f◦J(tj)∈Vjfor j= 1, . . . , J, so TJx∈V1× · · · × VJ×E×E× · · · ⊂ V. Consequently, TJ(U)∩V6=∅, as required. (c) This is due to the following facts: the projection π1is continuous and surjective (by (S3)), π1◦Πfn=fn◦π1and (Πf)◦n= Πf◦nfor all n∈N. 16 Now, we focus our attention on the searching of conditions on the function f that guarantee the universality of Bfand Πfon general SSSs. A local weakly mixing condition will be imposed in the following theorem on a sequence of selfmappings fixing the distinguished point. Theorem 4.3. Assume that fn:E→E(n∈N)is a sequence of selfmappings for which the mappings Tn:x= (xj)∈S7→ (fn(xj+n)) ∈Sare well-defined and continuous, where Sis a SSS on Esatisfying (S4*) for some distinguished point a∈E. Suppose that fn(a) = afor all n∈Nand that (fn)is weakly mixing at a. Then (Tn)is densely universal. Proof. Observe first that, in a similar way to the case of Bf, every fnmust be continuous, see Remark 3.3.1. Our aim is to apply Theorem 2.1. Choose X:= S=: Y. Observe that Xis Baire and that Yis second-countable by (S1). Consequently, our goal is, given a pair of nonempty open sets U, V of S, to find a sequence x= (xj)∈Uand a positive integer Nsuch that TNx∈V. From (S3), U∩σ(a)6=∅ 6=V∩σ(a). Therefore there exist J∈Nand points a1, . . . , aJ, b1, . . . , bJ∈Ewith α∈Uand β∈V, where α:= (a1, . . . , aJ, a, a, a, . . .) and β:= (b1, . . . , bJ, a, a, a, . . .). First of all, let us prove the following claim: There are in fact infinitely many N∈Nwith fN(A)∩Uj6=∅for all j= 1, . . . , m, where A, U1, . . . , Umare prescribed nonempty open sets with a∈A. Indeed, choose N1∈Nsuch that each fN1(A)∩Uj(j∈ {1, . . . , m}) is not empty. If Ehas only one point, namely a, then the claim is trivial. If Ehas at least two points then, since Eis Hausdorff, there are b∈Eand open subsets A0, B ⊂Ewith a∈A0, b ∈Band A0∩B=∅. Recall that each fnis continuous. Hence there exist open subsets An(n= 1, . . . , N1) in Ewith a∈Anand An⊂Asuch that fn(An)⊂A0; we have used that fn(a) = afor all n. Define e A:= A1∩. . . ∩AN1. Then e Ais an open subset containing the point aand fn(e A)⊂A0for all n∈ {1, . . . , N1}. In addition, e A⊂A. Hence fn(e A)∩B=∅for n= 1, . . . , N1. By hypothesis, there exists 17 N2∈N(necessarily, N2> N1) with fN2(e A)∩Uj6=∅for all j∈ {0,1, . . . , m}, where U0:= B. Therefore fN2(A)∩Uj6=∅(j∈ {1, . . . , m}), which proves the claim because in the same way we would obtain N1< N2< N3<· · · such that fNk(A)∩Uj6=∅for all j∈ {1, . . . , m}and all k∈N. Now we recover our first goal and fix U, V, α, β as before. Since α∈σJ(a)∩ U, from (S4*) it can be extracted the existence of finitely many open sets U1, . . . , UJ, A in Efor which α∈Π∞ j=1A(N) j⊂U(N > J), where A(N) jis defined by (1). In addition, there are open sets V1, . . . , VJin Esuch that β∈V1× · · · × VJ× {a}×{a} × · · · ⊂ V. By the just-proved claim, a positive integer Ncan be chosen in such a way that N > J and fN(A)∩Vj6=∅(j= 1, . . . , J). Hence there exist Jpoints t1, . . . , tJ,in Asatisfying fN(tj)∈Vj(j= 1, . . . , J). Let us define x= (xj)∈ENas xj=       αj(1 ≤j≤J) tj−N(N+ 1 ≤j≤N+J) a(J < j ≤Nor j > N +J). Then x∈Π∞ j=1A(N) j, so x∈U. Finally, TNx= (fN(xN+j)) = (t1, . . . , tJ, fN(a), fN(a), fN(a), . . .) = (t1, . . . , tJ, a, a, a, . . .)∈V1× · · · × VJ× {a}×{a}×{a} × · · · ⊂ V, which concludes the proof. Remark 4.4. The sufficient condition for universality furnished in the last theorem may not be necessary at all. Indeed, if for instance E=Rand S=ωthen the identity f:x∈R7→ x∈Rhas dense range and is continuous, so B=Bf 18 is universal by Theorem 4.2, but the sequence (fn)=(f◦n) is clearly not weakly mixing at a= 0. Nevertheless, the converse holds for c0and for the lpspaces. In fact, we will be able to obtain a more general result for SSSs (see Theorem 4.5 below) under the further condition that the “center” of Shas neighborhoods with projections which are as “uniformly small” as desired, that is, under the condition (S5) Given an open subset V⊂Econtaining a, there exists an open subset U⊂Swith (a, a, a, . . .)∈Usuch that πj(U)⊂Vfor all j∈N. For instance, c0(E) and lp(E) (0 <p<∞) satisfy (S5) (with a= 0) for any Banach space E, while ω(E) does not satisfy it for any metrizable space E. Theorem 4.5. If Sis a SSS on Esatisfying (S5) for the distinguished point a and fn:E→E(n∈N)is a sequence of selfmappings such that Tn:x= (xj)∈S7→ (fn(xj+n)) ∈S(n∈N)is a densely universal sequence of continuous selfmappings on S, then (fn)is weakly mixing at a. Proof. Let us fix an open set V⊂Econtaining the distinguished point a. Fix also finitely many nonempty open sets Uj⊂E(j= 1, . . . , J). Since (S5) holds for S, we get the existence of an open subset U⊂Scontaining (a, a, a, . . .) such that πj(U)⊂Vfor all j∈N. Since (Tn) is densely universal, there must be at least one element x= (xj)∈Uwhich is universal for (Tn). The set W:= U1×· · · ×UJ×E×E×· · · is open in Sby (S2), therefore there exists a positive integer Nsuch that TNx∈W, that is, (fN(xj+N)) ∈U1×· · ·× UJ×E×E×· · ·. In other words, fN(xj+N)∈Uj(j= 1, . . . , J). But every xj+Nbelongs to V, hence Uj∩fN(V)6=∅for 1 ≤j≤N, as required. Theorem 4.6. Let Sbe a SSS and Πfn:S→S(n∈N)be a sequence of Φ-product maps. We have: 19 (a) If (Πfn)is densely universal then, for all J∈N, the sequence {fn×· · ·×fn: EJ→EJ}n≥1is transitive. (b) If fn(a) = afor all n∈Nand the sequence {fn× · · · × fn:EJ→EJ}n≥1 is transitive for all J∈N, then (Πfn)is densely universal. Proof. (a) Fix J∈Nand nonempty open subsets A, B of EJ. We must show that (fN×· · ·×fN)(A)∩B6=∅for some N. There exist nonempty open subsets U1, . . . , UJ, V1, . . . , VJof Esuch that U1× · · · × UJ⊂Aand V1× · · · × VJ⊂B. Hence it is enough to find an Nwith fN(Uj)∩Vj6=∅for all j= 1, . . . , J. From (S2) the sets U:= (U1×· · ·×UJ×E×E×· · ·)∩Sand V:= (V1×· · ·×VJ×E× E× · · ·)∩Sare open in S. By hypothesis and Theorem 2.1 together with (S1), the sequence (Πfn) is transitive, so there exists N∈Nsuch that ΠfN(U)∩V6=∅. Pick an element y= (y1, y2, . . .) in such intersection. Then y∈Vand there is x= (x1, x2, . . .)∈Uwith fN(xj) = yjfor all j∈N. Hence xj∈Uj, yj∈Vjand fN(xj) = yj(j= 1, . . . , J), which proves (a). (b) This time we fix nonempty open subsets U, V of S. Again by Theorem 2.1 and (S1) it should be shown the existence of an Nwith ΠfN(U)∩V6=∅. Due to (S3), the sets σ(a)∩Uand σ(a)∩Vare nonempty, so σJ(a)∩U6=∅ 6=σJ(a)∩V for some J∈N. Now (S4) comes to our help, yielding the existence of nonempty open sets U1, . . . , UJ, V1, . . . , VJin Ewith U1× · · · × UJ× {a}×{a} × · · · ⊂ σJ(a)∩U and V1× · · · × VJ× {a}×{a} × · · · ⊂ σJ(a)∩V. By hypothesis, there exists N∈Nsuch that fN(Uj)∩Vj6=∅(j= 1, . . . , J). Pick points xj∈Uj,yj=fN(xj)∈Vj(j= 1, . . . , J). Then ex:= (x1, . . . , xJ, a, a, . . .)∈ σJ(a)∩U⊂Uand ey:= (y1, . . . , yJ, a, a, . . .)∈σJ(a)∩V⊂V. Finally, ΠfNex= (fN(x1), . . . , fN(xJ), fN(a), fN(a), . . .)=(y1, . . . , yJ, a, a, . . .) = ey because every fnfixes a. 20 We remark that in Theorem 4.6(a) only properties (S1)–(S2) of an SSS are used; in particular, it also holds for the space ⊕n∈NK, see Example 3.2.2. Roughly speaking, the following corollary shows that under soft conditions on a SSS the universality of the backward Φ-shifts and of the Φ-product maps becomes completely characterized in terms of the dynamical properties of their underlying selfmappings. Corollary 4.7. We have the following: (a) Assume that Bf:S→Sis a Φ-shift on a SSS Ssatisfying (S4*) with distinguished point a, in such a way that f(a) = aand fis weakly mixing at that point. Then Bfis universal. (b) Assume that Bf:S→Sis a universal Φ-shift on a SSS Swhich satisfies property (S5). Then fis weakly mixing at the distinguished point. (c) Assume that Πf:S→Sis a Φ-product map on a SSS Swith distinguished point a, in such a way that f(a) = aand fis weakly mixing. Then Πfis universal. (d) Assume that Πf:S→Sis a universal Φ-product map on a SSS. Then f is weakly mixing. (e) Suppose that Eis a separable Banach space, that S=lp(E)or c0(E) (0 < p < ∞)and that f:E→Eis continuous. In addition, we assume lim supt→0kf(t)k/ktk<∞if S=lp(E), and f(0) = 0 if S=c0(E). Then the Φ-product Πf(the Φ-shift Bf, resp.) is universal on Sif and only if f is weakly mixing (weakly mixing at the origin, resp.). Proof. The results (a)–(e) are direct consequences of Theorems 3.5, 4.3, 4.5, 4.6 and of the fact that, for a single selfmapping Ton a topological space, the universality of Timplies the dense universality of the sequence (T◦n) of its iterates. Only part (c) needs some further explanation: Since fis weakly mixing, by [11, Proposition II.3] one gets that for every J∈Nthe mapping 21 f×· · ·×f:EJ→EJis transitive. But this is the same as the transitivity of the sequence f◦n× · · · × f◦n:EJ→EJ(n∈N), hence Theorem 4.6(b) applies. 5 Applications to hypercyclicity theory Here we obtain two examples of hypercyclic operators and of hypercyclic sequences of operators on linear SSSs as a consequence of some preceding results. Firstly, we get the following rather general statement that extends Rolewicz’s theorem. This is just the case E=K,S=lp(1 ≤p < ∞) or c0,f= the identity on K. Theorem 5.1. Let be prescribed a Banach space E, a surjective operator fon Eand a linear SSS Son Esatisfying (S4*) for a= 0. Let UEbe the open unit ball of E. Then the scalar multiple λBf:S→Sof the backward Φ-shift Bfis hypercyclic whenever |λ|> µ := 1 sup{α > 0 : f(UE)⊃αUE}. . Proof. The Open Mapping Theorem together with the boundedness of fguarantees that µ∈(0,∞). If |λ|> µ then f(UE)⊃αUEfor some α∈(0,∞) with |λα|>1. Therefore (λf)◦n(UE)⊃(λα)nUEfor all n∈N. Let us fix an open set V⊂Ewith 0 ∈Vand finitely many nonempty open sets U1, . . . , UJin E. Pick points tj∈Uj(j= 1, . . . , J). Then V⊃βUEfor some β > 0. Since the set F:= {t1, . . . , tJ}is finite and |λα|>1, there is N∈Nsuch that β(λα)NUE⊃F. Hence, trivially, (λf)◦N(V)∩Uj6=∅for every j∈ {1, . . . , J}. Then λf is weakly mixing at the origin, so Corollary 4.7(a) applies with a= 0 if we take into account that Bλf =λBf. We finish with a result (Theorem 5.2) that relates the hypercyclicity of a Φproduct map to the so-called Hypercyclicity Criterion, which is the condition (b) 22 in Theorem 5.2. Such criterion is a well known sufficient condition for hypercyclicity, see [7], [14] and [5]. We will assume that Sis a complete linear SSS on an Fspace E. Hence Sis a separable F-space (due to (S1), because second-countable is equivalent to metrizable plus separable) and, from (S4), Eis also separable. The following concept was introduced by the author in [4]: a sequence (fn) of operators on Eis called almost-commuting whenever limn→∞[fn(fm(t))−fm(fn(t))] = 0 for every m∈Nand every t∈E. Theorem 5.2. Suppose that Sis a complete linear SSS on an F-space E. Assume that (fn)is a sequence of operators on Esuch that (Πfn)defines a sequence of operators on S. Consider the following properties: (a) The sequence (Πfn)is densely hypercyclic. (b) There exist dense subsets X0and Y0of Eand an increasing sequence (nk)⊂ Nsatisfying the following two conditions: fnk(t)→0 (k→ ∞)for all t∈X0; for any t∈Y0there is a sequence (uk)in Esuch that uk→0and fnk(uk)→t(k→ ∞). (c) The sequence fn×fn:E2→E2(n∈N)is hypercyclic. Then we have the following: (A) Properties (a) and (b) are equivalent. (B) If (fn)is almost-commuting then (a), (b) and (c) are equivalent. Proof. (A) We are assuming that (Πfn) is densely hypercyclic. From Theorem 4.6(a), the sequence {fn× · · · × fn:EJ→EJ}n≥1is transitive (so densely hypercyclic by Theorem 2.1) for all J∈N. Then Theorem 2.2 of [5] applies and one obtains that (b) holds. Conversely, assume that (b) is satisfied. Again by Theorem 2.2 of [5] the sequence {fn× · · · × fn:EJ→EJ}n≥1is densely hypercyclic (so transitive) for all J∈N. Since fn(0) = 0 for all nwe have that (Πfn) is densely hypercyclic by Theorem 4.6(b). (B) We have already obtained that (a) and (b) are equivalent. On the other hand, if (a) holds then by Theorem 4.6(a) (for J= 2) we get again the transitivity 23 (so the dense hypercyclicity, hence the single hypercyclicity) of the sequence {fn×fn:E2→E2}n≥1. Conversely, if this sequence is hypercyclic then Theorem 3.3 of [5] guarantees that (b) is satisfied. Of course, part (B) applies to a single operator fon Esince any two iterates f◦n, f◦mclearly commute. Other conditions on (fn) which are equivalent to the Hypercyclicity Criterion can be seen in [5] and [7]. Observe also that the transitivity of {fn× · · · × fn:EJ→EJ}n≥1for all J∈Nis stronger than the property that (fn) is weakly mixing at the origin. Hence, in view of Theorem 4.3 and of the proof of Theorem 5.2, we have that the sequence {Tn:x= (xj)∈ S7→ (fn(xj+n)) ∈S}n≥1is hypercyclic if (b) and (S4*) (with a= 0) are satisfied. In particular if fis an operator on Ethen under the latter two conditions (with fn=f◦nfor all n∈N) the Φ-shift Bfis hypercyclic on S. Remark 5.3. As for a nice nonlinear example, we point out that some arguments similar to those presented in the proofs of Theorems 4.3 and 4.5 allowed Peris to show in [25] that a polynomial P:lq→lqgiven by P(x1, x2, . . .) = (p(x2), p(x3), . . .) –where p:C→Cis a complex polynomial with p(0) = 0 of degree strictly greater that one– is universal if and only if 0 belongs to the Julia set of p. ACKNOWLEDGEMENT The author is grateful to A. Peris and the referee for helpful comments and suggestions. References [1] J. Banks, Topological mapping properties defined by digraphs, Discrete and Cont. Dyn. Syst. 5(1999), 83–92. [2] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney’s definition of chaos, Amer. Math. Monthly 99 (1992), 332–334. [3] L. Bernal-Gonz´alez, Universal functions for Taylor shifts, Complex Variables Theory Appl. 31 (1996), 121–129. 24 [4] L. Bernal-Gonz´alez, Universal images of universal elements, Studia Math. 138 (2000), 241–250. [5] L. Bernal-Gonz´alez and K.G. Grosse-Erdmann, The Hypercyclicity Criterion for sequences of operators, Studia Math. 157 (2003), 17–32. [6] N. Bernardes, On orbits of polynomial maps in Banach spaces, Quaestiones Math. 21 (1998), 311–318. [7] J.P. B`es and A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999), 94–113. [8] J. Bonet, F. Mart´ınez-Gim´enez and A. Peris, Linear chaos on Fr´echet spaces, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), 1649–1655. [9] R.L. Devaney, An introduction to Chaotic Dynamical Systems, AddisonWesley, Reading, 1989. [10] P.L. Duren, Theory of Hpspaces, Academic Press, New York, 1970. [11] H. Furstenberg, Disjointness in Ergodic Theory, minimal sets and a problem in diophantine approximation, Systems Theory 1(1967), 1–49. [12] G. Godefroy and J.H. Shapiro, Operators with dense invariant cyclic vector manifolds, J. Funct. Anal. 98 (1991), 229–269. [13] K.G. Grosse-Erdmann, Holomorphe Monster und universelle Funktionen, Mitt. Math. Sem. Giessen 176 (1987), 1–84. [14] K.G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. 36 (1999), 345–381. [15] K.G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math. 139 (2000), 47–68. [16] K.G. Grosse-Erdmann, Recent developments in hypercyclicity, Rev. R. Acad. Cien. Serie A. Mat. 97 (2003), 273–286. [17] G. K¨othe, Topological vector spaces, I, Springer-Verlag, Berlin/New York, 1969. 25