Alfredo Peris

K.-G. Grosse-Erdmann

Grosse-Erdmann, K.-G.

Peris, Alfredo

Numérisation de Documents Anciens Mathématiques

1/Spainquently. Math.rove thatart etépond àBayart etorbite quiC. R. Acad. Sci. Paris, Ser. I 341 (2005) 123–128http://france.elsevier.com/direct/CRASSDynamical SystemsFrequently dense orbitsK.-G. Grosse-Erdmanna,1, Alfredo Perisb,2a Fachbereich Mathematik, FernUniversität Hagen, 58084 Hagen, Germanyb E.T.S. Arquitectura, Departament de Matemàtica Aplicada and IMPA-UPV, Universitat Politècnica de València, 46022 València,Received 21 May 2005; accepted 24 May 2005Available online 1 July 2005Presented by Gilles PisierAbstractWe study the notion of frequent hypercyclicity that was recently introduced by Bayart and Grivaux. We show that frehypercyclic operators satisfy the Hypercyclicity Criterion, answering a question of Bayart and Grivaux [Trans. AmerSoc., in press]. We also disprove a conjecture therein concerning frequently hypercyclic weighted shifts, and we pvectors which have a somewhere frequently dense orbit are frequently hypercyclic.To ite this article: K.-G. Grosse-Erdmann,A. Peris, C. R. Acad. Sci. Paris, Ser. I 341 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméOrbites fréquemment denses.On étudie la notion d’hypercyclicité fréquente qui a récemment été introduite par BayGrivaux. Nous démontrons que tout opérateur fréquemment hypercyclique vérifie le Critère d’Hypercyclicité, ce qui rune question de Bayart et Grivaux [Trans. Amer. Math. Soc., à paraître]. De plus, nous réfutons une conjecture deGrivaux concernant les shifts à poids fréquemment hypercycliques, et nous démontrons que tout vecteur avec uneest quelque part fréquemment dense est fréquemment hypercyclique.Pour citer cet article : K.-G. Grosse-Erdmann, A. Peris,C. R. Acad. Sci. Paris, Ser. I 341 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.E-mail addresses:kg.grosse-erdmann@fernuni-hagen.de (K.-G. Grosse-Erdmann), aperis@mat.upv.es (A. Peris).1 Partially supported by a grant from Programa Incentivo a la Investigación of the Universitat Politècnica de València.2 Partially supported by MEC and FEDER, Project MTM2004-02262 and by AVCIT Grupos 03/050.1631-073X/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2005.05.025124 K.-G. Grosse-Erdmann, A. Peris / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 123–128lan-vide.euCritèreyclicitéetteque partmulti-lmmenttyucedVersion française abrégéeUn opérateurT sur un espace vectoriel topologique séparableX st dithypercycliques’il existe un vecteurx ∈ X tel que l’orbite{T nx: n = 1,2,3, . . .} est dense dansX. Bayart et Grivaux [1,2] ont récemment introduitnotion d’hypercyclicité fréquente en quantifiant la fréquence avec laquelle l’orbite rencontre tout ouvert noDéfinition 0.1. Un opérateurT sur un espace vectoriel topologique séparableX est ditfréquemment hypercycliqus’il existe un vecteurx ∈ X tel que, pour tout ouvert non-videU deX,dens{n ∈ N: T nx ∈ U} > 0,où, pour toute partieA de N, dens(A) = lim infN→∞ 1N card{n  N : n ∈ A}. Un tel vecteur est appelévecteurfréquemment hypercycliquepourT .Bayart et Grivaux [2, Question 4.8] demandent si, pour tout opérateur fréquemment hypercycliqueT , ’opérateurT ⊕ T est hypercyclique. D’après un résultat de Bès et Peris [4], cette dernière propriété deT est équivalente aCritère d’Hypercyclicité. Par conséquent, le théorème suivant implique une réponse positive.Théorème 0.2.Tout opérateur fréquemment hypercyclique sur un espace métrisable complet vérifie led’Hypercyclicité.Dans la Conjecture 2.10 de [2], Bayart et Grivaux proposent une condition pour caractériser l’hypercfréquente d’un shift à poidsT surp, 1  p < ∞, donné parT (xn) = (wn+1xn+1) avec des poids(wn) positifs.Nous donnons une condition nécessaire pour queT soit fréquemment hypercyclique, et nous prouvons que ccondition est strictement plus forte que celle de la Conjecture 2.10, ce qui réfute la conjecture.De plus, nous donnons l’analogue du résultat de Bourdon et Feldman [6] que toute orbite qui est queldense est partout dense.Théorème 0.3. Soit T un opérateur sur un espace vectoriel topologiqueX. S’il existe un vecteurx ∈ X et unouvert non-videU deX tel quedens{n ∈ N: T nx ∈ V } > 0 pour toute partie ouverte non-videV deU , alorsx estfréquemment hypercyclique pourT .Comme corollaire on obtient l’analogue du résultat de Costakis [7] et Peris [9] que tout opérateurhypercyclique est hypercyclique.Théorème 0.4.SoitT un opérateur sur un espace vectoriel topologiqueX. S’il existe des vecteursx1, . . . , xN ∈ Xtels que, pour tout ouvert non-videU de X, dens⋃Nj=1{n ∈ N: T nxj ∈ U} > 0, alors un des vecteursxj , j =1, . . . ,N, est fréquemment hypercyclique pourT .Ce dernier résultat implique que, pour tout opérateur fréquemment hypercycliqueT s r un espace vectorietopologique, l’opérateurT N,N ∈ N, est aussi fréquemment hypercyclique, avec les mêmes vecteurs fréquehypercycliques queT , cf. [2, Theorem 4.7].1. IntroductionA (continuous and linear) operatorT on a separable topological vector spaceX is calledhypercyclicif thereexists a vectorx ∈ X whose orbit{T nx: n = 1,2,3, . . .} is dense inX, that is, if its orbit meets every non-empopen subsetU of X. Motivated by Birkhoff’s ergodic theorem, Bayart and Grivaux [1,2] have recently introdK.-G. Grosse-Erdmann, A. Peris / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 123–128 125r [5]. Insolvingon andere densegue fore.secondthe notion of a frequently hypercyclic operator by quantifying the frequency with which an orbit meetsU . Werecall that thelower densityof a subsetA of N is defined asdens(A) = lim infN→∞card{n  N : n ∈ A}N.Definition 1.1. An operatorT on a separable topological vector spaceX is calledfrequently hypercyclicif thereexists a vectorx ∈ X such that, for every non-empty open subsetU of X, the set{n ∈ N: T nx ∈ U}has positive lower density. Each such vectorx is then calledfrequently hypercyclicfor T .This new concept has been thoroughly studied by Bayart and Grivaux [2] and Bonilla and the first authothis note we show that every frequently hypercyclic operator satisfies the Hypercyclicity Criterion, therebyQuestion 4.8 of [2], and we give a counterexample to Conjecture 2.10 of [2].In our proofs we shall need the following well-known result, cf. [10]. The definition ofupper densityfor subsetsof N is analogous to that of lower density.Theorem 1.2.If A ⊂ N has positive upper density then the difference setD := A−A = {m−n: m,n ∈ A,m > n}has bounded gaps, that is, there is someM ∈ N such thatD ∩ [n,n + M] = ∅ for all n ∈ N.We begin, however, with a remarkable general property of frequent hypercyclicity. By a theorem of BourdFeldman [6], an operator on a separable locally convex space is hypercyclic as soon as it has a somewhorbit; in fact, the assumption of local convexity may be dropped, cf. Wengenroth [11]. We obtain the analofrequent hypercyclicity.Theorem 1.3.Let T be an operator on a topological vector spaceX. If there is a vectorx ∈ X and a non-emptyopen subsetU of X such thatdens{n ∈ N: T nx ∈ V } > 0 for all non-empty open subsetsV of U , thenx isfrequently hypercyclic forT .Proof. It follows from the theorem of Bourdon and Feldman thatT is hypercyclic, hence topologically transitivThus, if U ′ is an arbitrary non-empty open subset ofX there is someN ∈ N such thatT N(U) ∩ U ′ = ∅. Hencethere is a non-empty open subsetV of U such thatT N(V ) ⊂ U ′, whence dens{n ∈ N: T nx ∈ U ′}  dens{N + n:T nx ∈ V } > 0, so thatx is frequently hypercyclic forT . As a consequence we obtain the analogue for frequent hypercyclicity of the result of Costakis [7] and theauthor [9] that every multi-hypercyclic operator is hypercyclic. The proof is by induction onN .Theorem 1.4.Let T be an operator on a topological vector spaceX. If there are vectorsx1, . . . , xN ∈ X suchthat, for any non-empty open subsetU of X, dens⋃Nj=1{n ∈ N: T nxj ∈ U} > 0, then somexj , j = 1, . . . ,N, isfrequently hypercyclic forT .This result, in turn, implies that for any frequently hypercyclic operatorT on a topological vector spaceX theoperatorT N,N ∈ N, is also frequently hypercyclic, and it has the same frequently hypercyclic vectors asT . Thisgeneralizes Theorem 4.7 of [2].126 K.-G. Grosse-Erdmann, A. Peris / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 123–128e, andon forr thentfies theperator, see [4].swer tont. Interion.fied.-2. The Hypercyclicity CriterionIn this section we letX denote an F-space, that is, a complete and metrizable topological vector spacwe assume thatX is separable. In that case the so-called Hypercyclicity Criterion gives a sufficient conditian operator onX to be hypercyclic, see [4] or [8]. For the purpose of this note it will be enough to considecriterion in one of its equivalent forms.Theorem 2.1[3,4]. LetT be an operator on a separable F-spaceX. Then the following assertions are equivale:(i) T satisfies the Hypercyclicity Criterion;(ii) the operatorT ⊕ T is hypercyclic;(iii) for any non-empty open subsetsU andV of X and for any neighbourhoodW of 0 in X there exists an ∈ Nsuch thatT n(U) ∩ W = ∅ andT n(W) ∩ V = ∅.One of the main problems in the theory of hypercyclic operators asks if every hypercyclic operator satisHypercyclicity Criterion. It seems to be accepted wisdom in hypercyclicity that the answer is positive if the ohas some additional regularity, like, for example, a dense set of periodic points (so that it becomes chaotic)However, frequent hypercyclicity seems to impose rather more irregularity than regularity; hence the anQuestion 4.8 of [2] if every frequently hypercyclic operator satisfies the Hypercyclicity Criterion is not evidefact, the answer turns out to be in the affirmative.Theorem 2.2.Every frequently hypercyclic operator on a separable F-space satisfies the Hypercyclicity CriProof. Suppose thatT is frequently hypercyclic. We want to show that condition (iii) of Theorem 2.1 is satisThus, letU andV be non-empty open subsets ofX andW a neighbourhood of 0 inX.Since T is necessarily hypercyclic, hence topologically transitive, there is someN ∈ N such thatT N(U) ∩ W = ∅. Then there is a non-empty open subsetU0 of U such thatT N(U0) ⊂ W . Moreover, if x isfrequently hypercyclic forT there isA ⊂ N of positive lower density such thatT nx ∈ U0 for all n ∈ A, henceT N+m−nT nx ∈ T N(U0) ⊂ W for all m,n ∈ A with m > n. LettingD = A − A we deduce thatT k(U) ∩ W = ∅ for all k ∈ N + D. (1)By Theorem 1.2 the setN + D has bounded gaps, so that there is someM ∈ N with(N + D) ∩ [n,n + M] = ∅ for all n ∈ N. (2)Now, sinceW is a neighbourhood of 0 there exists a 0-neighbourhoodW0 such thatT n(W0) ⊂ W for n =0,1, . . . ,M . Again by topological transitivity ofT , there exists someK > M and somew0 ∈ W0 with T Kw0 ∈ V ,henceT K−nT nw0 ∈ V for n = 0,1, . . . ,K . Therefore we haveT k(W) ∩ V = ∅ for all k ∈ [K − M,K]. (3)Thus, (1), (2) and (3) imply that there exists somek ∈ N with T k(U) ∩ W = ∅ andT k(W) ∩ V = ∅, which hadto be shown. We have thus shown that for any frequently hypercyclic operatorT the operatorT ⊕ T is hypercyclic. Question 4.9 of [2] if T ⊕ T is even frequently hypercyclic, however, remains open.K.-G. Grosse-Erdmann, A. Peris / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 123–128 127ndsaryty ofr,3. Weighted backward shiftsAn operatorT on the sequence spacep,1  p < ∞, is a weighted backward shift if there is a positive, anecessarily bounded, sequence(wn) such thatT (xn) = (wn+1xn+1) for (xn)n0 ∈ p. In [2] it is shown that∞∑n=11(w1 · · ·wn)p < ∞ (4)is a sufficient condition for frequent hypercyclicity ofT , see also [5], while the following is obtained as a necescondition: there exists a sequence(nk) of positive lower density such that∞∑k=11(w1 · · ·wnk )p< ∞. (5)In Conjecture 2.10 of [2] it is suggested that the latter condition, in fact, characterizes frequent hypercycliciT .We show here that this is not the case; to this end we derive a stronger necessary condition.Proposition 3.1.If a weighted backward shiftT with weights(wn) is frequently hypercyclic onp then, for everyε > 0, there exists a sequence(nk) of positive lower density such that, for alli ∈ N,∞∑k=i+11(w1 · · ·wnk−ni )p< ε.Proof. Let x ∈ p be frequently hypercyclic forT and 0< η < 1. Then there exists a sequence(nk) of positivelower density such that‖T nkx − e0‖ < η for all k ∈ N, wheree0 = (1,0,0, . . .). This implies that∣∣(w1 · · ·wnk )xnk − 1∣∣ < η,hence(w1 · · ·wnk )|xnk | > 1− η for all k ∈ N. (6)Moreover, we have for alli ∈ Nηp > ‖T ni x − e0‖p ∞∑j=1(w1 · · ·wj+niw1 · · ·wj)p|xj+ni |p ∞∑k=i+1(w1 · · ·wnkw1 · · ·wnk−ni)p|xnk |p,hence by (6)∞∑k=i+11(w1 · · ·wnk−ni )p<ηp(1− η)p ,which implies the result. Example 1. There exists a weighted backward shift onp with limn→∞ wn = 1 that satisfies condition (5) fosome sequence(nk) of positive lower density but that is not frequently hypercyclic.Proof. Let Nj = 2(j − 1)j for j ∈ N. We definewn = (n+1)2n2 for Nj < n  Nj + j , wn = (Nj + j + 1)−2/j forNj + j < n  Nj +2j , wn = 1 for Nj +2j < n  Nj +3j andwn = (Nj+1 +1)2/j for Nj +3j < n  Nj +4j =Nj+1, wherej ∈ N. Then limn→∞ wn = 1; and we havew1 · · ·wn = (n + 1)2 for all n ∈ ⋃j1[Nj + 1,Nj + j ],so that condition (5) is satisfied for this set of positive lower density. Now, ifT were frequently hypercyclic then128 K.-G. Grosse-Erdmann, A. Peris / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 123–128sty, norersitators are8.7–32..by Proposition 3.1, there would exist a sequence(mk) of positive lower density such thatw1 · · ·wmk−mi > 1 for allk > i  1. On the other hand, we havew1 · · ·wn = 1 for Nj + 2j < n  Nj + 3j , hence forj consecutive valueof n, wherej is arbitrarily large. By Theorem 1.2 this is impossible.We do not know if the necessary condition of Proposition 3.1 is also sufficient for frequent hypercycliciif condition (4) is necessary.AcknowledgementsThis work was done while the first author visited the Department of Applied Mathematics at the UnivPolitècnica de València. He wishes to thank the department for their kind hospitality. Moreover, the authgrateful to Frédéric Bayart and Sophie Grivaux for sharing with them their paper [2].References[1] F. Bayart, S. Grivaux, Hypercyclicité : le rôle du spectre ponctuel unimodulaire, C. R. Acad. Sci. Paris, Ser. I 338 (2004) 703–70[2] F. Bayart, S. Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc., in press.[3] L. Bernal-González, K.-G. Grosse-Erdmann, The hypercyclicity criterion for sequences of operators, Studia Math. 157 (2003) 1[4] J. Bès, A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999) 94–112.[5] A. Bonilla, K.-G. Grosse-Erdmann, Frequently hypercyclic operators, Preprint.[6] P.S. Bourdon, N.S. Feldman, Somewhere dense orbits are everywhere dense, Indiana Univ. Math. J. 52 (2003) 811–819.[7] G. Costakis, On a conjecture of D. Herrero concerning hypercyclic operators, C. R. Acad. Sci. Paris, Sér. I 330 (2000) 179–182[8] K.-G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999) 345–381.[9] A. Peris, Multi-hypercyclic operators are hypercyclic, Math. Z. 236 (2001) 779–786.[10] C.L. Stewart, R. Tijdeman, On infinite-difference sets, Canad. J. Math. 31 (1979) 897–910.[11] J. Wengenroth, Hypercyclic operators on non-locally convex spaces, Proc. Amer. Math. Soc. 131 (2003) 1759–1761.

Frequently dense orbits

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Frequently dense orbits

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