DYNAMICS OF TUPLES OF MATRICES

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volue 37, Nuber 3, Mrch 2009, Pges S Article electroniclly published on October 7, 2008 DYNAMICS OF TUPLES OF MATRICES G. COSTAKIS, D. HADJILOUCAS, AND A. MANOUSSOS Counicted by Nigel J. Klton Abstrct. In this rticle we nswer question rised by N. Feldn in 2008 concerning the dynics of tuples of opertors on R n. In prticulr, we prove tht for every positive integer n 2 there exist n-tuples A,A 2,...,A n of n n trices over R such tht A,A 2,...,A n is hypercyclic. We lso estblish relted results for tuples of 2 2 trices over R or C being in Jordn for.. Introduction Following the recent wor of Feldn in [4] n n-tuple of opertors is finite sequence of length n of couting continuous liner opertors T,T 2,...,T n cting on loclly convex spce X. The tuple T,T 2,...,T n is hypercyclic if there exists vector x X such tht the set {T T 2 2 T n n x :, 2,..., n 0 is dense in X. Such vector x is clled hypercyclic for T,T 2,...,T n ndtheset of hypercyclic vectors for T,T 2,...,T n will be denoted by HCT,T 2,...,T n. The bove definition generlizes the notion of hypercyclicity to tuples of opertors. For n ccount of results, coents nd n extensive bibliogrphy on hypercyclicity we refer to [], [5], [6] nd [7]. For results concerning the dynics of tuples of opertors see [2], [3], [4] nd [9]. In [4] Feldn showed, ong other things, tht in C n there exist digonlizble n + -tuples of trices hving dense orbits. In ddition he proved tht there is no n-tuple of digonlizble trices on R n or C n tht hs soewhere dense orbit. Therefore the following question rose nturlly. Question Feldn [4]. Are there non-digonlizble n-tuples on R tht hve soewhere dense orbits? We give positive nswer to this question in very strong for, s the next theore shows. Received by the editors Mrch 24, Mthetics Subject Clssifiction. Priry 47A6. Key words nd phrses. Hypercyclic opertors, tuples of trices. During this reserch the third uthor ws fully supported by SFB 70 Spetrle Struturen und Topologische Methoden in der Mtheti t the University of Bielefeld, Gerny. He would lso lie to express his grtitude to Professor H. Abels for his support. 025 c 2008 Aericn Mtheticl Society

2 026 G. COSTAKIS, D. HADJILOUCAS, AND A. MANOUSSOS Theore.. For every positive integer n 2 there exist n-tuples A,...,A n of n n non-siultneously digonlizble trices over R such tht A,...,A n is hypercyclic. Restricting ourselves to tuples of 2 2 trices in Jordn for either on R 2 or C 2, we prove the following. Theore.2. There exist 2 2 trices A j,j =, 2, 3, 4, in Jordn for over R such tht A,A 2,A 3,A 4 is hypercyclic. In prticulr { x HCA,A 2,A 3,A 4 = R 2 : x x Theore.3. There exist 2 2 trices A j,j =, 2,...,8, in Jordn for over C such tht A,A 2,...,A 8 is hypercyclic. 2. Products of 2 2 trices Le 2.. Let be positive integer nd for ech j =, 2,..., let A j be j 2 2 trix in Jordn for over field F = C or R, i.e. A j = for 0 j, 2,..., F. ThenA,A 2,...,A over C respectively R is hypercyclic if nd only if the sequence { :, 2,..., N is dense in C 2 respectively R 2. Proof. We prove the bove in the cse F = C, since the other cse is siilr. Observe tht A l l l j l j = j l 0 j for l N. Asresultwehve j j j s j s A A A = j= j= s= 0 j. j Assue tht A,A 2,...,A is hypercyclic nd let z z 2 C 2 be hypercyclic vector for A,A 2,...,A. Then the sequence { A A A z : z, 2,..., N 2 = j j + z 2 z j= z 2 j j j= j j j= s= s s j= :, 2,..., N is dense in C 2. This iplies tht z 2 0. Dividing the eleent in the first row by tht in the second, it cn esily be shown tht the sequence { :, 2,..., N is dense in C 2. The converse cn esily be shown.

3 DYNAMICS OF TUPLES OF MATRICES 027 Rer 2.2. Let be positive integer nd for ech j =, 2,..., let A j be 2 2 trix in Jordn for over field F = C or R. By the proof of Le 2. it is iedite tht whenever A,A 2,...,A overc respectively R is hypercyclic, one cn copletely describe the set of hypercyclic vectors s { z z 2 C 2 : z 2 0 respectively { x x 2 R 2 : x The rel cse. The one-diensionl version of Kronecer s theore stted below see for exple [8, Theore 438, p. 375] will be used repetedly throughout this wor. Theore 2.3. If x is positive irrtionl nuber, then the sequence {x s :, s N is dense in R. Rer 2.4. If x is positive irrtionl nuber, then the sequence {s x :, s N is lso dense in R. Liewise, if x is negtive irrtionl nuber, then the sequence {s + x :, s N is dense in R. We shll need the following well-nown result; see for exple [4]. Theore 2.5. If, b > nd ln ln b is dense in R +. is irrtionl, then the sequence { n b : n, N Le 2.6. Let, b R such tht <<0, b> nd Then the sequence { n b : n, N is dense in R. ln ln b is irrtionl. ln Proof. Since ln b is irrtionl it follows tht ln b/ ln is irrtionl s well. Applying Theore 2.5 we conclude tht the sequence { 2n b : n, N is dense 2 in R +. On the other hnd the fct tht is negtive iplies tht the sequence { 2n+ b : n, N is dense in R. This copletes the proof of the le. Proposition 2.7. There exist, 2, 3, 4 R such tht the sequence { :, 2, 3, 4 N 4 is dense in R 2. Proof. Bythelebovefix, b R such tht <<0, + R \ Q nd { n b : n, N is dense in R. Let x,x 2 R nd ɛ>0 be given. Then there exist n, N such tht n b x 2 <ɛ.notetht n b = n+ b s for every, s N. Notelsotht + < 0. Hence, by Rer 2.4, the sequence { s + + :, s N is dense in R; i.e. there exist, s N such tht s + + x n <ɛ, b i.e. n + b s x <ɛ. Hence, setting =, 2 = b, 3 =, 4 = we prove the result.

4 028 G. COSTAKIS, D. HADJILOUCAS, AND A. MANOUSSOS Proof of Theore.2. This is n iedite consequence of Le 2., Proposition 2.7 nd Rer 2.2. Exple 2.8. One y construct ny concrete exples of four 2 2 trices, in Jordn for over R, being hypercyclic. For exple, fix, b R such tht <<0, b>ndboth + ln, ln b re irrtionl. Fro the bove we conclude tht b,,, 0 0 b 0 0 is hypercyclic. We shll now prove Theore. for n = 2; see Proposition 2.0 ii. For this we need the following result due to Feldn; see Corollry 3.2 in [4]. Proposition 2.9 Feldn. Let D denote the open unit dis centered t 0 in the coplex plne. If b D \{0, then there exists dense set C \ D such tht for every the sequence { n b : n, N is dense in C. Proposition 2.0. i Every pir A,A 2 of 2 2 trices over R with A j, j =, 2, being either digonl or in Jordn for is not hypercyclic. ii There exist pirs A,A 2 of 2 2 trices over R such tht A is digonl, A 2 is ntisyetric rottion trix nd A,A 2 is hypercyclic. In prticulr every non-zero vector in R 2 is hypercyclic for A,A 2 ; i.e. HCA,A 2 = R 2 \{0, 0. iii There exist pirs A,A 2 of 2 2 trices over R such tht both A nd A 2 re ntisyetric nd A,A 2 is hypercyclic. In prticulr every non-zero vector in R 2 is hypercyclic for A,A 2, i.e. HCA,A 2 = R 2 \{0, 0. Proof. Let us prove ssertion i. The cse of A,A 2 both digonl is covered by Feldn; see [4]. Assue tht A is digonl nd A 2 is in Jordn for; i.e. A = 0 0, A 2 = b 0 b for, b R. Suppose tht A,A 2 is hypercyclic nd let x x 2 R 2 be hypercyclic vector for A,A 2. Then the sequence { A n A 2 x x 2 : n, N = { n b x + n b x 2 n b : n, N x 2 is dense in R 2. Therefore b cnnot be zero. Observe tht x 2 cnnot be zero either. Te ny y R nd y 2 R \{0. Then there exist sequences of positive integers {n, { such tht + nd n b x + n b x 2 y, n b x 2 y 2 s +. Sinceb 0,y 2 0ndx 2 0wegettht n b x y 2x x 2 nd n b x 2 = b n b x 2 +

5 DYNAMICS OF TUPLES OF MATRICES 029 s +. Fro the lst, it clerly follows tht n b x + n b x 2 +, which is contrdiction. Assue now tht both A,A 2 re in Jordn for; i.e. b A =, A 0 2 =, 0 b for, b R nd A,A 2 is hypercyclic. Le 2. iplies tht the sequence { n + b n b : n, N is dense in R 2. Observe tht neither nor b is equl to. By ting the bsolute vlue in the second coordinte nd then pplying the logrithic function, we find tht the sequence { is dense in R 2. Hence the sequence { n + b n ln + ln b ln n n ln ln + b + ln b : n, N : n, N is dense in R 2. Subtrcting the second coordinte fro the first one, we conclude tht the sequence { ln ln b : N b is dense in R, which is bsurd. We proceed with the proof of ssertion ii. By Proposition 2.9 there exist R \ Q nd b C such tht the sequence { n b : n, N is dense in C. Writeb = b e iθ nd set 0 b cos θ b sin θ A =, A 0 2 =. b sin θ b cos θ Then we hve A n A 2 = n b cos θ n b sin θ n b sin θ n b cos θ Applying in the bove reltion the vector 0 nd ting into ccount tht the sequence { n b : n, N is dense in C, we conclude tht the sequence { A n A 2 0 : n, N = { n b cos θ n b sin θ. : n, N is dense in R 2. Hence A,A 2 is hypercyclic. It is now esy to show tht every non-zero vector in R 2 is hypercyclic for A,A 2. In order to prove the lst ssertion we follow siilr line of resoning s bove. Tht is, by Proposition 2.9 there exist, b C such tht the sequence { n b : n, N is dense in C. Write = e iφ, b = b e iθ nd set cos φ sin φ b cos θ b sin θ A = sin φ cos φ, A 2 = b sin θ b cos θ.

6 030 G. COSTAKIS, D. HADJILOUCAS, AND A. MANOUSSOS A direct coputtion gives tht { A n A 2 0 { n b cosnφ + θ n b sinnφ + θ : n, N is equl to : n, N, nd by the choice of, b we conclude tht the vector 0 is hypercyclic for A,A 2. This copletes the proof of the proposition. Question 2.. Wht is the iniu nuber of 2 2 trices over R in Jordn for so tht their tuple fors hypercyclic opertor? 2.2. The coplex cse. In wht follows we will be writing Rz nd Iz for the rel nd iginry prts of coplex nuber z respectively. Proposition 2.2. There exist j C, j =, 2,...,8 such tht the sequence { :, 2,..., 8 N 8 is dense in C 2. Proof. The proof is in the se spirit s the proof of Proposition 2.7. Fix, b C such tht <<0, +, R \ Q nd {n b : n, N is dense in C see Proposition 2.9. Let z,z 2 C nd ɛ>0 be given. Then there exist n, N such tht n b z 2 <ɛ.notetht n b = n+ b s i ξ i ξ 4i ρ ρ 4 for every, s, ξ N nd ρ 4N. Note tht + < 0nd > 0. Hence, by Theore 2.3, the sequence { ξ ρ : ξ N,ρ 4N 4 is dense in R. As result, there exist ξ N nd ρ 4N such tht iξ I ρ i I z n 4 <ɛ; b i.e. we hve tht n I + b + iξ ρ i Iz 4 <ɛ. By Rer 2.4, the sequence { + + s :, s N is dense in R. Hence, there exist, s N such tht + + s 4ρ + R z n <ɛ; b i.e. we hve tht R n + b + + 4ρ + s Rz <ɛ.

7 DYNAMICS OF TUPLES OF MATRICES 03 But this ens tht the rel nd iginry prts of the coplex nuber n + b s + iξ i ρ 4 4ρ re within ɛ of the rel nd iginry prts of z. Hence, setting =, 2 = b, 3 =, 4 =, 5 = i, 6 = i, 7 =4i, 8 = 4,weprovetheresult. Proof of Theore.3. By Proposition 2.2, Le 2. nd Rer 2.2 the ssertion follows. Exple 2.3. Fix, b C such tht <<0, +, R \ Q nd { n b : n, N is dense in C. Fro the bove it is evident tht the 8-tuple of 2 2 trices in Jordn for over C given by b,, 0 0 b 0 i, i, 0 i 0 i 4i 0 4i, 0,, is hypercyclic. Question 2.4. Wht is the iniu nuber of 2 2 trices over C in Jordn for so tht their tuple fors hypercyclic opertor? 3. Products of 3 3 trices In this section we strt with the following specil cse of Corollry 3.5 in [4], due to Feldn, which will be of use to us in the following. Proposition 3. Feldn. If b,b 2 D \{0, then there exists dense set C \ D such tht for every, 2 the sequence { n b n l : n,, l N 2 b 2 is dense in C 2. In order to hndle products of 3 3 trices, we estblish the following: Corollry 3.2. There exist C nd b, c, d R such tht the sequence { n b c n d l : n,, l N is dense in C R. Proof. Fix two rel nubers b,b 2 with b,b 2 0,. By Proposition 3. there exist, 2 C \ D such tht the sequence { n b n l : n,, l N 2 b 2 is dense in C 2. Define =, b = b, c = 2 nd d = b 2. Observe tht the sequence { n b c n l : n,, l N b 2 is dense in C [0, +. Te z C nd x R.

8 032 G. COSTAKIS, D. HADJILOUCAS, AND A. MANOUSSOS Cse I. x 0. Then there exist sequences of positive integers {n, {, {l such tht n b z nd c n l b 2 x. l Since b 2 = d 2l we get c n d 2l x. Cse II. x<0. Then there exist sequences of positive integers {n, {, {l such tht n b z nd c n l b 2 x d. The lst iplies tht c n d 2l + x. This copletes the proof of the corollry. The in result of this section is to prove Theore. for n = 3. This is stted ndprovedbelow. Proposition 3.3. There exist 3 tuples A,A 2,A 3 of 3 3 trices over R such tht A,A 2,A 3 is hypercyclic. Proof. By Corollry 3.2 there exist C nd b, c, d R such tht the sequence { n b c n d l : n,, l N is dense in C R. Write = e iθ nd set cos θ sin θ 0 A = sin θ cos θ 0,A 2 = b b 0 nd 0 0 c 0 0 A 3 = d Then we hve A n A 2 A l 3 = whichinturngives A n A l 2 A 3 0 = n b cos nθ n b sin nθ 0 n b sin nθ n b cos nθ c n d l n b cos nθ n b sin nθ c n d l., The lst nd the choice of, b, c, d iply tht A,A 2,A 3 is hypercyclic with being hypercyclic vector for A,A 2,A 3. 0

9 DYNAMICS OF TUPLES OF MATRICES Proof of Theore. By Proposition 2.0, there exist 2 2 trices B nd B 2 such tht B,B 2 is hypercyclic. Cse I. n = 2 for soe positive integer. For = the result follows by Proposition 2.0. Assue tht >. Ech A j will be constructed by blocs of 2 2 trices. Let I 2 be the 2 2 identity trix. We will be using the nottion digd,d 2,...,D n to denote the digonl trix with digonl entries the bloc trices D,D 2,...,D n. Define A = digb,i 2,...,I 2, A 2 = digb 2,I 2,...,I 2, A 3 = digi 2,B,I 2,...,I 2, A 4 = digi 2,B 2,I 2,...,I 2 nd so on up to A n = digi 2,...,I 2,B, A n = digi 2,...,I 2,B 2. It is now esy to chec tht A,A 2,...,A n is hypercyclic nd furtherore tht the set HCA,A 2,...,A n is {x,x 2,...,x n R n : x 2 2j + x 2 2j 0, j =, 2,...,. Cse II. n = 2 + for soe positive integer. If = the result follows by Proposition 3.3. Suppose >. For siplicity we tret the cse = 2, since the generl cse follows by siilr rguents. By Proposition 3.3 there exist C,C 2,C 3,3 3 trices such tht C,C 2,C 3 is hypercyclic. Let I 3 be the 3 3 identity trix. Define A = digb,i 3, A 2 = digb 2,I 3, A 3 = digi 2,C, A 4 = digi 2,C 2 nda 5 = digi 2,C 3. ItcnesilybeshownthtA,A 2,...,A 5 is hypercyclic. The detils re left to the reder. Acnowledgeent The uthors would lie to thn the nonyous referee for helpful nd constructive coents. References [] F. Byrt, É. Mtheron, Topics in liner dynics, Cbridge Trcts in Mth. to pper. [2] N. S. Feldn, Hypercyclic Tuples of Opertors. Mini-worshop: Hypercyclicity nd Liner Chos. Abstrcts fro the ini-worshop held August 3-9, Orgnized by Teres Berúdez, Gilles Godefroy, Krl-G. Grosse-Erdnn nd Alfredo Peris. Oberwolfch Reports,Vol.3,no.3.OberwolfchRep , no. 3, MR [3] N.S.Feldn,Hypercyclic pirs of conlytic Toeplitz opertors, Integrl Equtions Opertor Theory , MR d:47057 [4] N.S.Feldn,Hypercyclic tuples of opertors nd soewhere dense orbits, J. Mth. Anl. Appl , [5] K.-G. Grosse-Erdnn, Universl filies nd hypercyclic opertors, Bull. Aer. Mth. Soc , MR c:4700 [6] K.-G. Grosse-Erdnn, Recent developents in hypercyclicity, RACSAM Rev. R. Acd. Cienc. Excts Fis. Nt. Ser. A Mt , MR c:4700 [7] K.-G. Grosse-Erdnn, Dynics of liner opertors, Topics in coplex nlysis nd opertor theory, 4-84, Univ. Mlg, Mlg, MR

10 034 G. COSTAKIS, D. HADJILOUCAS, AND A. MANOUSSOS [8] G. H. Hrdy nd E. M. Wright, An Introduction to the Theory of Nubers, 5th edition, The Clrendon Press, Oxford University Press, New Yor, 979. MR i:0002 [9] L. Kérchy, Cyclic properties nd stbility of couting power bounded opertors, Act Sci. Mth. Szeged , MR e:4704 Deprtent of Mthetics, University of Crete, Knossos Avenue, GR Herlion, Crete, Greece E-il ddress: costis@th.uoc.gr Deprtent of Coputer Science nd Engineering, Europen University Cyprus, 6 Diogenes Street, Engoi, P.O. Box 22006, 56 Nicosi, Cyprus E-il ddress: d.hdjiloucs@euc.c.cy Fultät für Mtheti, SFB 70, Universität Bielefeld, Postfch 003, D-3350 Bielefeld, Gerny E-il ddress: nouss@th.uni-bielefeld.de