Dynamics of tuples of matrices in Jordan form
aa r X i v : . [ m a t h . F A ] D ec . DYNAMICS OF TUPLES OF MATRICES IN JORDAN FORM
GEORGE COSTAKIS AND IOANNIS PARISSIS (1) A bstract . A tuple ( T , . . . , T k ) of n × n matrices over R is called hypercyclic if for some x ∈ R n the set { T m T m · · · T m k k x : m , m , . . . , m k ∈ N } is dense in R n . We prove that the minimumnumber of n × n matrices in Jordan form over R which form a hypercyclic tuple is n +
1. I ntroduction
Let X be a separable Banach space either over R or C . Recall that a bounded linearoperator T : X → X is hypercyclic if there exists a vector x ∈ X whose orbit Orb( T , x ) = { x , Tx , T x , . . . } is dense in X . For a thorough study of hypercyclicity we refer to the recentbook (Bayart and Matheron, 2009). Although hypercyclicity is a phenomenon which onlyappears in infinite dimensions, see (Kitai, 1982), Feldman recently established that this isnot the case if one considers more than one operator; see (Feldman, 2008).Following Feldman from (Feldman, 2008), we give the following definition:1.1. Definition.
Let T = ( T , . . . , T k ) be a k -tuple of commuting continuous linear operators,acting on X . The k -tuple T will be called hypercyclic if there exists a vector x ∈ X such thatthe set { T m x : m ∈ N k } = { T m T m · · · T m k k x : m , m , . . . , m k ∈ N } , is dense in X . Here we use the standard multi-index notation where m = ( m , . . . , m k ) ∈ N k and T m = T m . . . T m k k .Here and throughout the paper N denotes the set of positive integers while N denotesthe set of non-negative integers, N = N ∪ { } .Specializing to the case X = R n or X = C n we have that T is a k -tuple of commuting n × n matrices over R or C respectively. In (Feldman, 2008), Feldman proved that in C n there exist Mathematics Subject Classification.
Primary: 47A16 Secondary: 11J72, 15A21.
Key words and phrases. hypercyclic operator, Jordan form, Kronecker’s theorem. (1)
Partially supported by CAMGSD-LARSYS through Fundac¸ ˜ao para a Ciˆencia e Tecnologia(FCT / Portugal), program POCTI / FEDER..
G. COSTAKIS AND I. PARISSIS ( n + simultaneously diagonalizable matrices which are hypercyclic. Furthermore,Feldman proved that there is no hypercyclic n -tuple of simultaneously diagonalizable matricesacting on R n or C n . On the other hand, in (Costakis et al., 2009), the authors proved that on R n , n ≥
2, there exist n -tuples of non-simultaneously diagonalizable matrices over R whichare hypercyclic. For further results on hypercyclic tuples of operators in finite or infinitedimensions look at (Feldman, 2006),(Feldman, 2007), (K´erchy, 2005), (Javaheri, 2009) and(Costakis et al., 2010).In this note we restrict our attention to k -tuples of non-simultaneously diagonalizablematrices on R n , n ≥
2, where every operator in the k -tuple T is in Jordan form. In generalwe will write Jrd l ,γ for the Jordan block of dimension l with eigenvalue γ , that is:Jrd l ,γ ≔ γ . . . γ . . . ... γ . . . ... . . . . . . . . . . . . γ , where γ ∈ R . A general n × n matrix in Jordan form over R , considered here, will consistof p Jordan blocks and in general can be represented as J = diag { Jrd n ,γ , Jrd n ,γ , . . . , Jrd n p ,γ p } = Jrd n ,γ ⊕ Jrd n ,γ ⊕ · · · ⊕ Jrd n p ,γ p , where n + · · · + n p = n and γ , . . . , γ p ∈ R .A few remarks are in order:1.2. Remark.
Throughout the exposition we consider k -tuples T = ( T , . . . , T k ) of n × n matrices in Jordan form over R . In general each T ν , 1 ≤ ν ≤ k , will have a di ff erent numberof Jordan blocks of di ff erent dimensions. However, since we consider k -tuples of operatorsthat commute , it is not hard to see that all the matrices in the k -tuple must have the sameform, that is, it is enough to consider the case that all the k operators T ν have p Jordanblocks of dimensions n , . . . , n p , where p and n , . . . , n p do not depend on which term ν inthe k -tuple we are considering. Bearing this in mind, each operator T ν can be written inthe form(1.3) T ν = Jrd n ,γ (1) ν ⊕ · · · ⊕ Jrd n p ,γ ( p ) ν , where the number of blocks p and the corresponding dimensions n , . . . , n p are fixedthroughout the k -tuple. Thus, the real number γ ( b ) ν is the eigenvalue of the b -th Jordanblock in the ν -th operator of the k -tuple.1.4. Remark.
Suppose for a moment that n = · · · = n p =
1, in other words, that all the Jordanblocks in the k -tuple are of dimension one. Because of Remark 1.2, this means that all theoperators in the k -tuple are diagonal. However, this case has already been considered by YNAMICS OF TUPLES OF MATRICES IN JORDAN FORM 3
Feldman in (Feldman, 2008). We will therefore assume for the rest of the paper that n b > k -tuple.Concerning Jordan forms in R , the following Theorem was proved in (Costakis et al., 2009)1.5. Theorem (Costakis, Hadjiloucas, Manoussos (Costakis et al., 2009)) . There exist × matrices A j , j = , , , in Jordan form over R such that ( A , A , A , A ) is hypercyclic. The authors in (Costakis et al., 2009) raised the following Question:1.6.
Question.
What is the minimum number of × matrices in Jordan form over R so that theirtuple is hypercyclic? In Theorem 1.5, since the dimension is two, all the matrices in Jordan form are necessarilyJordan blocks, that is each matrix has a single eigenvalue (remember we exclude the casethat one of the matrices is diagonal). It is not hard to see that the conclusion of Theorem1.5 is exceptional as in R n , for n ≥
3, there is no k -tuple of matrices, each one being exactlya Jordan block, which is hypercyclic:1.7. Proposition.
Let n ≥ and k ∈ N . For any γ , . . . , γ k ∈ R consider the k-tuple of Jordanblocks J = (Jrd n ,γ , Jrd n ,γ , . . . , Jrd n ,γ k ) . Then J is not hypercyclic. Thus in dimension n ≥ k -tuples T = ( T , . . . , T k ) where each one ofthe matrices T ν , 1 ≤ ν ≤ k , is in Jordan form over R and consists of more than one Jordanblocks.For n × n matrices in Jordan form over R we have the following result in the negativedirection:1.8. Proposition.
For n , k ∈ N we consider a k-tuple of n × n matrices in Jordan form over R ,T = ( T , . . . , T k ) , where each T ν consists of p Jordan blocks of dimensions n , . . . , n p as in (1.3) .(i) Suppose that n b ≥ for at least one b ∈ { , , . . . , p } . Then T is not hypercyclic.(ii) If k = n then T is not hypercyclic. Remark.
Observe that part (ii) of Proposition 1.8 is only interesting when all the Jordanblocks in each one of the matrices of the n -tuple have dimension n b ≤
2. Otherwise, part(i) gives a stronger statement.The main result of this paper is the following theorem:1.10.
Theorem.
Fix a positive integer n ≥ and let p ≥ and p ≥ be given non-negativeintegers such that p + p = n. There exists a hypercyclic ( n + -tuple of n × n matrices in Jordanform over R where each matrix in the tuple consists of p Jordan blocks of dimension and p Jordan blocks of dimension . G. COSTAKIS AND I. PARISSIS
Remark.
The case n = R n for any n ≥ Corollary.
The minimum number of n × n matrices in Jordan form over R which form ahypercyclic tuple is n + . The rest of the paper is organized as follows. In section 3 we present some generalguidelines and conventions concerning the notations in this paper. In section 4 we presentsome calculations which occur frequently in dealing with Jordan blocks. We are able toturn the hypercyclicity condition in a condition which is linear in ( m , . . . , m k ). This willturn out to be much more flexible than the original definition of hypercyclicity. In section5 we take advantage of this linear reformulation of the definition of hypercyclicity in orderto prove the negative results contained in Propositions 1.7 and 1.8.Finally, section 6 contains the proof of the main result, Theorem 1.10. The proof ofTheorem 1.10 relies on the linear reformulation of the problem mentioned before. Inparticular, we need to construct a matrix L + such that the set { L + m T : m ∈ N n + } is dense in R n . The entries of L + have a special structure, imposed by the fact that we consider tuples ofmatrices in Jordan form over R . In Theorem 6.8, we exploit the multi-dimensional version ofKronecker’s theorem in order to reduce the construction of the matrix L + to the constructionof a certain set of vectors, the rows of L + , which should be linearly independent over Q .The construction of these vectors is done by induction in the dimension n in conjunctionwith the solution of a non-linear equation in Lemma 6.9 which guarantees that the entriesof our vectors will have the desired structure. We will take up all these issues in the finalsection of this paper. 2. A cknowledgements We would like to thank the anonymous referee for an expert reading and suggestionsthat helped us improve the quality of this paper.3. N otations
A few words about the notation are necessary. In many parts of the paper the notationbecomes cumbersome due to the nature of the problem. However we consistently use thesame notation which we present now. In general we will consider k -tuples T = ( T , . . . , T k ) YNAMICS OF TUPLES OF MATRICES IN JORDAN FORM 5 of n × n matrices in Jordan form over R . The ν -th matrix in the k -tuple consists of p Jordanblocks of dimensions n , . . . , n p with n + · · · + n p = n . Each block is in turn defined bymeans of its dimension and a real eigenvalue. We will always use the symbol b , where1 ≤ b ≤ p , to index the blocks. Thus a typical operator in the k -tuple is of the form T ν = Jrd n ,γ (1) ν ⊕ · · · ⊕ Jrd n p ,γ ( p ) ν = ⊕ pb = Jrd n b ,γ ( b ) ν . Hopefully these general guidelines will help the reader throughout the exposition.4. A uxiliary C alculations All the results in this note depend heavily on some explicit calculations. There are twotypes of calculations involved in the proof. The first concerns operations on single Jordanblocks, that is, powers of Jordan blocks and multiplication of powers of Jordan blocks.Such calculations appear for example in the context of Proposition 1.7. The second typeof calculations concerns operations on matrices that consist of several Jordan blocks each.However, since every matrix in the k -tuple is block diagonal, all the operations we areconsidering here go through in each block as in the case of single Jordan blocks; each blockbehaves independently than the other blocks in terms of taking powers and multiplyingwith other Jordan matrices in the k -tuple, since all matrices have the same block structure.4.1. Operations on single Jordan blocks.
First we calculate the powers of a single Jordanblock:4.1.
Lemma.
For n , m ∈ N and γ ∈ R \ { } we have (Jrd n ,γ ) m = γ m p ( m )1 p ( m )2 . . . p ( m ) n − p ( m )1 . . . ... . . . p ( m )2 ... . . . . . . . . . p ( m )1 . . . . The real numbers p ( m ) j are defined asp ( m ) j = mj ! γ j , j = , . . . , n − , with the understanding that (cid:0) mj (cid:1) = whenever m < j. In the next Lemma we calculate the product of powers of Jordan blocks.
G. COSTAKIS AND I. PARISSIS
Lemma.
Let n , k ∈ N and m = ( m , . . . , m k ) ∈ N k . Let γ , . . . , γ k be the eigenvalues thatdefine the Jordan blocks and set γ = ( γ , . . . , γ k ) . We also set J = (Jrd n ,γ , . . . , Jrd n ,γ k ) . We thenhave J m = γ m d ( m )1 d ( m )2 . . . d ( m ) n − d ( m )1 . . . ... . . . d ( m )2 ... . . . . . . . . . d ( m )1 . . . , where now the diagonals of J m are defined by the numbers d ( m ) j :d ( m ) j = X | β | = j m β ! γ β = X β + ··· + β k = j ≤ β p ≤ j , p = ,..., k m β ! · · · m k β k ! γ β · · · γ β k , j = , . . . , n − . The next step is to express the entries d ( m )1 , d ( m )2 in the first two diagonals in a simplerform. This will be enough for our purposes here. For d ( m )1 we readily see that d ( m )1 = k X ν = m ν γ ν . (4.3)For d ( m )2 we have d ( m )2 = k X ν = m ν ( m ν − γ ν + X ≤ ν<ν ′ ≤ k m ν m ν ′ γ ν γ ν ′ = (cid:18) ( d ( m )1 ) − k X ν = m ν γ ν (cid:19) . (4.4)4.2. Operations on matrices with several Jordan blocks.
We now consider the generalcase where we have a k -tuple of n × n matrices in Jordan form over R . As we have pointedout, the structure of each matrix should be the same for the operators to be commuting,that is, each matrix in the k -tuple consists of say p Jordan blocks with dimensions n , . . . , n p ,where n + · · · + n p = n . To fix the notation, let T = ( T , . . . , T k ). We have that T ν = Jrd n ,γ (1) ν ⊕ · · · ⊕ Jrd n p ,γ ( p ) ν , ≤ ν ≤ k , where γ (1) ν , . . . , γ ( p ) ν ∈ R for all 1 ≤ ν ≤ k . In other words, each matrix in the k -tuple is a blockdiagonal Jordan matrix with p discrete real eigenvalues. For b ∈ { , , . . . , p } , the blockJrd n b ,γ ( b ) ν is a n b × n b Jordan block with eigenvalue γ ( b ) ν , where the index ν tells us which termof the k -tuple we are considering. We may have n b = b ’s but we exclude the YNAMICS OF TUPLES OF MATRICES IN JORDAN FORM 7 possibility that n b = b ∈ { , , . . . , p } , that is, we don’t allow a k -tuple of diagonalmatrices. For m = ( m , . . . , m k ) ∈ N k , we can write T m = T m · · · T m k k = Π ⊕ · · · ⊕ Π p , (4.5)where each Π b is the tuple defined as Π b def = Jrd m n b ,γ ( b )1 · · · Jrd m k n b ,γ ( b ) k , b ∈ { , , . . . , p } . (4.6)Observe that each block Π b is described by Lemma 4.2 with n = n b and γ = ( γ ( b )1 , . . . , γ ( b ) k ).4.3. Matrices with Jordan blocks of dimension at most two.
We now turn our attentionto k -tuples of n × n matrices in Jordan form over R where all the Jordan blocks havedimension n b ≤
2. This is justified because of Proposition 1.8 which says that if even one ofthe Jordan blocks has dimension n b > k -tuple cannot be hypercyclic. To simplifythe notation let us agree that in each Jordan matrix of the k -tuple we have p Jordan blocksof dimension 2 and p Jordan blocks of dimension 1. Assume that in the ν -th term the 2 × γ (1) ν , . . . , γ ( p ) ν and the 1 × c (1) ν , . . . , c ( p ) ν . We also write γ ( b ) = ( γ ( b )1 , . . . , γ ( b ) k ) for 1 ≤ b ≤ p and c ( b ) = ( c ( b )1 , . . . , c ( b ) k ) for 1 ≤ b ≤ p . We define the matrix Γ = { γ ( b ) ν } ∈ R p × k whose rows arethe vectors γ ( b ) , 1 ≤ b ≤ p . Similarly, the matrix C = { c ( b ) ν } ∈ R p × k is the matrix whose rowsare the vectors c ( b ) for 1 ≤ b ≤ p . Of course we have 2 p + p = n . Finally, the followingnotation will be useful. For Γ and C as before and m = ( m , . . . , m k ) ∈ N k , we consider thevector in R n : V ( m , Γ , C ) def = (cid:18) ( γ (1) ) m , k X ν = m ν γ (1) ν , . . . , ( γ ( p ) ) m , k X ν = m ν γ ( p ) ν , ( c (1) ) m , . . . , ( c ( p ) ) m (cid:19) . Let T = ( T , . . . , T k ) be a k -tuple of n × n matrices in Jordan form over R , satisfying theprevious assumptions. For m = ( m , . . . , m k ) ∈ N k , T m will be given by a form similar to(4.5): T m = T m · · · T m k k = P ⊕ · · · ⊕ P p ⊕ P ′ ⊕ . . . ⊕ P ′ p . The tuples P b are defined as: P b def = Jrd m ,γ ( b )1 · · · Jrd m k ,γ ( b ) k , b ∈ { , , . . . , p } , while P ′ b def = Jrd m , c ( b )1 · · · Jrd m k , c ( b ) k = ( c ( b ) ) m , b ∈ { , , . . . , p } . G. COSTAKIS AND I. PARISSIS
With these notations and assumptions taken as understood, we use Lemma 4.2 togetherwith the expression (4.3) to get a more handy characterization of hypercyclicity in thespecial case we are considering.4.7.
Lemma.
Let T = ( T , . . . , T k ) be a k-tuple of n × n matrices in Jordan form over R , defined bymeans of the matrices Γ and C. We assume that all the Jordan blocks in T have dimension at mosttwo. Then T is hypercyclic if and only if the set (cid:26) V ( m , Γ , C ) : m = ( m , . . . , m k ) ∈ N k (cid:27) , is dense in R n .Proof. Let T = ( T , . . . , T k ) be a k -tuple of n × n matrices over R where each matrix in thetuple has p Jordan blocks of dimension 2 and p Jordan blocks of dimension 1. For any m ∈ N k and y ∈ R n a straightforward calculation using Lemma 4.2 yields( T m y ) T = ( γ (1) ) m (cid:16) y + P k ν = m ν γ (1) ν y (cid:17) ( γ (1) ) m y ... ( γ ( p ) ) m (cid:16) y p − + P k ν = m ν γ ( p ν y p (cid:17) ( γ ( p ) ) m y p ( c (1) ) m y p + ... ( c ( p ) ) m y p + p . (4.8)Assume now that T is a hypercyclic tuple. There exists y = ( y , . . . , y n ) ∈ R n such that { T m y : m ∈ N k } = R n . (4.9)By (4.8) and (4.9) we conclude that y j , j ∈ { , , , . . . , p , p + , p + , . . . , p + p } .Now let x ∈ R n be a vector with all of its entries di ff erent than zero. We define the vector z = ( z , . . . , z n ) ∈ R n by defining its coordinates: z b − = x b y b − + x b − x b y b , if b ∈ { , , . . . , p } , z b def = x b y b , if b ∈ { , , . . . , p } , z p + b def = x p + b y p + b , if b ∈ { , , . . . , p } . Since the k -tuple T is hypercyclic, there exists a sequence { m ( τ ) } τ ∈ N ⊂ N k such that T m ( τ ) y → z in R n as τ → + ∞ . By the definition of the vector z and (4.8) we getlim τ → + ∞ ( γ ( b ) ) m ( τ ) = x b for all b = , , . . . , p , YNAMICS OF TUPLES OF MATRICES IN JORDAN FORM 9 lim τ → + ∞ ( c ( b ) ) m ( τ ) = x p + b for all b = , , . . . , p , and lim τ → + ∞ ( γ ( b ) ) m ( τ ) ( y b − + k X ν = m ( τ ) n γ ( b ) ν y b ) = x b y b − + x b − x b y b , for all b = , , . . . , p . Combining the previous convergence relations we conclude thatlim τ → + ∞ k X ν = m ( τ ) n γ ( b ) ν = x b − for all b = , , . . . , p . Observe that in order to conclude the previous results we had to divide by entries of x or y but this is justified since we have made sure that these entries are non-zero.We have showed that if T is hypercyclic then for every x ∈ R n with all of its entriesdi ff erent than zero there exists a sequence { m ( τ ) } τ ∈ N ⊂ N k such that V ( m ( τ ) , Γ , C ) → x as τ → + ∞ . Since the set { x = ( x , x , . . . , x n ) ∈ R n : x j , j = , , . . . n } is dense in R n , this concludes one direction of the equivalence in the lemma.The opposite directions is very easy. Choose w ∈ R n with w = w = · · · = w p − = w = w = · · · = w p = w p + = · · · = w n =
1. If x ∈ R n has all of its entries di ff erent thanzero we chose { m ( τ ) } τ ∈ N ⊂ N k such that( γ ( b ) ) m ( τ ) → x b as τ → + ∞ for all b = , , . . . , p , k X ν = m ( τ ) n γ ( b ) ν → x b − / x b as τ → + ∞ for all b = , , . . . , p , ( c ( b ) ) m ( τ ) → x b as τ → + ∞ for all b = , , . . . , p . This is always possible by our hypothesis. Using (4.8) it is easy to see that T m ( τ ) w → x as τ → + ∞ . It readily follows that w is a hypercyclic vector for T . (cid:3) A slight variant helps us write this in linear form in terms of m ∈ N k . Corollary.
Let T = ( T , . . . , T k ) be a k-tuple of n × n matrices in Jordan form over R whereall the Jordan blocks have dimension at most two. We define the n × k matrixL = log | γ (1)1 | log | γ (1)2 | · · · log | γ (1) k | /γ (1)1 /γ (1)2 · · · /γ (1) k ... ... · · · ... log | γ ( p )1 | log | γ ( p )2 | · · · log | γ ( p ) k | /γ ( p )1 /γ ( p )2 · · · /γ ( p ) k log | c (1)1 | log | c (1)2 | · · · log | c (1) k | ... ... · · · ... log | c ( p )1 | log | c ( p )2 | · · · log | c ( p ) k | . If T is hypercyclic then the set { Lm T : m ∈ N k } is dense in R n . Here m T denotes the transpose of the vector m = ( m , . . . , m k ). Proof.
Let us denote by V + ( m , Γ , C ) the vector V + ( m , Γ , C ) def = (cid:18) | ( γ (1) ) m | , k X ν = m ν γ (1) ν , . . . , | ( γ ( p ) ) m | , k X ν = m ν γ ( p ) ν , | ( c (1) ) m | , . . . , | ( c ( p ) ) m | (cid:19) . Since T is hypercyclic, Lemma 4.7 implies that the set (cid:26) V ( m , Γ , C ) : m = ( m , . . . , m k ) ∈ N k (cid:27) is dense in R n and, thus, that the set (cid:26) V + ( m , Γ , C ) : m = ( m , . . . , m k ) ∈ N k (cid:27) is dense in ( R + × R ) p × ( R + ) p . Now let x ∈ R n and define the vector y def = ( e x , x , e x , x , . . . , e x p − , x p , e x p + , e x p + . . . , e x p + p ) . Since y ∈ ( R + × R ) p × ( R + ) p there exists a sequence { m ( τ ) } τ ∈ N ⊂ N k such that V + ( m ( τ ) , Γ , C ) → y as τ → + ∞ . This convergence is equivalent to | γ ( b )1 | m ( τ )1 · · · | γ ( b ) k | m ( τ ) k → e x b − as τ → + ∞ for all b = , , . . . , p , m ( τ )1 γ ( b )1 + · · · + m ( τ ) k γ ( b ) k → x b as τ → + ∞ for all b = , , . . . , p , | c ( b )1 | m ( τ )1 · · · | c ( b ) k | m ( τ ) k → e x p + b as τ → + ∞ for all b = , , . . . , p . YNAMICS OF TUPLES OF MATRICES IN JORDAN FORM 11
Taking logarithms, the previous three convergence relations are equivalent to m ( τ )1 log | γ ( b )1 | + · · · + m ( τ ) k log | γ ( b ) k | → x b − as τ → + ∞ for all b = , , . . . , p , m ( τ )1 γ ( b )1 + · · · + m ( τ ) k γ ( b ) k → x b as τ → + ∞ for all b = , , . . . , p , (4.11) m ( τ )1 log | c ( b )1 | + · · · + m ( τ ) k log | c ( b ) k | → x p + b as τ → + ∞ for all b = , , . . . , p . Since x ∈ R n was arbitrary, gathering the convergence relations (4.11) in matrix formgives L ( m ( τ ) ) T → x as τ → + ∞ . Since x ∈ R n was arbitrary this concludes the proof of thelemma. (cid:3)
5. N egative R esults In this section we prove Propositions 1.7 and 1.8
Proof of Proposition 1.7.
Let n , k ∈ N and γ = ( γ , . . . , γ k ) ∈ R k be the eigenvalues defininga k -tuple of Jordan blocks J = ( J n ,γ , . . . , J n ,γ k ). Now suppose J is hypercyclic, that is, thereexists a x ∈ R n such that the set { J m x : m ∈ N k } is dense in R n .Let J ( m ) be the 3 × J m that arises from J m by deleting the first n − n − J ( m ) = γ m d ( m )1 d ( m )2 d ( m )1 . Since { J m x : m ∈ N k } is dense in R n there exists a y = ( y , y , y ) ∈ R such that the set { J ( m ) y : m ∈ N k } is dense in R . In particular the set { γ m y : m ∈ N k } is dense in R so we must have y ,
0. Now we let w = ( y + y + y , y + y , y ) and we choose asequence m = m ( τ ) = ( m ( τ )1 , m ( τ )2 , m ( τ )3 ) such that J m ( τ ) y → w as τ → ∞ . We will suppress τ tosimplify notation. Since y , γ m → τ → ∞ . Next we have that γ m ( y + y d ( m )1 ) → y + y as τ → ∞ . We conclude that d ( m )1 → τ → ∞ . Finally, from thefirst row of J m we get that γ m ( y + d ( m )1 y + d ( m )2 y ) → y + y + y as τ → ∞ . Recalling theformula for d ( m )2 in equation (4.4) we can rewrite this as γ m (cid:16) y + d ( m )1 y +
12 (( d ( m )1 ) − k X j = m j γ j ) y (cid:17) → y + y + y as τ → ∞ . (5.1) Let us write ℓ = lim τ →∞ P kj = m j γ j which obviously exists. From (5.1) we then get that y + y + y − ℓ y = y + y + y . But this means that ℓ = − P kj = m j γ j ≥ m ∈ N k . (cid:3) We now give the proof of the more general result for k -tuples of n × n matrices in Jordanform over R . Proof of Proposition 1.8.
Let us assume that a k -tuple T of matrices in Jordan form over R ishypercyclic. For m ∈ N k the matrix T m has the form given by (4.5) and (4.6). For (i) let Π b be the block that has dimension n = n b ≥
3. Let this block be defined by the real numbers γ ( b )1 , . . . , γ ( b ) k . Fixing this b , we just write γ = ( γ , . . . , γ k ). Equation (4.6) shows that Π b willbe of the form Π b = Jrd m n b ,γ · · · Jrd m k n b ,γ k . Sine T is hypercyclic and T m is a block diagonal matrix, we conclude that there exists a y ∈ R n b such that { Π b y : m ∈ N k } is dense in R n b . Since n b ≥ n -tuples of n × n matrices in Jordan form, T = ( T , . . . , T n ), where each one of the matrices T ν consists of p Jordan blocks of dimension n b ≤ b ∈ { , , . . . , p } . We adopt the notations from paragraph 4.3. Using Corollary4.10 we see that if T is hypercyclic then { Lm T : m = ( m , . . . , m n ) ∈ N n } = R n . But this meansthat the operator L : R n → R n has dense range and therefore is onto. We conclude that L is invertible so we must have N n = R n , a contradiction. (cid:3) Remark.
In part (i) of Proposition 1.8 we show that if at least one of the Jordan blocksin the tuple has dimension n b ≥ k -tuple is hypercyclic. However, the proof givenabove works equally well to give a stronger statement, namely that the tuple T is not even somewhere dense : for every x ∈ R n , the closure of the set { T m x : m ∈ N k } does not contain anyopen balls.5.3. Remark.
Likewise, the proof of part (ii) of Proposition 1.8 gives the stronger statementthat an n -tuple of n × n matrices in Jordan form over R is never somewhere dense. Indeed,if the orbit of the n -tuple T is somewhere dense for some x in R n then there is a ball B inside the set L ( R n ). Then the set L ( R n ), which is a linear subspace of R n , has necessarilydimension n . We conclude that L ( R n ) = R n so that the matrix L is invertible and then weproceed as in the proof above. YNAMICS OF TUPLES OF MATRICES IN JORDAN FORM 13
6. H ypercyclic tuples of matrices in J ordan form In this section we give the proof of Theorem 1.10. For this we need to construct ( n + n × n matrices in Jordan form over R which are hypercyclic. For technical reasonswe need to consider the two-dimensional case separately than the n -dimensional case for n ≥
3. We first give the proof in R n for n ≥ The proof in the case n ≥ . We recall that each matrix in the tuple we want toconstruct will consist of p Jordan blocks of dimension 2 and p blocks of dimension 1.Thus we necessarily have 2 p + p = n . Since n ≥ p , p ∈ N we conclude that p + p ≥
2. We consider the vectors γ (1) def = (cid:16) − a (1)1 , a (1)2 , a (1)3 , . . . , a (1) p − , a (1) p , a (1) p + , . . . , a (1) n , − a (1) n + (cid:17) ,γ (2) def = (cid:16) a (2)1 , − a (2)2 , a (2)3 , . . . , a (2) p − , a (2) p , a (2) p + , . . . , a (2) n , − a (2) n + (cid:17) ,... (6.1) γ ( p −
1) def = (cid:16) a ( p − , a ( p − , a ( p − , . . . , − a ( p − p − , a ( p − p , a ( p − p + , . . . , a ( p − n , − a ( p − n + (cid:17) ,γ ( p ) def = (cid:16) a ( p )1 , a ( p )2 , a ( p )3 , . . . , a ( p ) p − , − a ( p ) p , a ( p ) p + , . . . , a ( p ) n , − a ( p ) n + (cid:17) , where a ( b ) ν > ≤ ν ≤ n + ≤ b ≤ p . Similarly let us define c (1) def = (cid:16) − δ (1)1 , δ (1)2 , δ (1)3 , . . . , δ (1) p − , δ (1) p , δ (1) p + , . . . , δ (1) n + (cid:17) , c (2) def = (cid:16) δ (2)1 , − δ (2)2 , δ (2)3 , . . . , δ (2) p − , δ (2) p , δ (2) p + , . . . , δ (2) n + (cid:17) ,... (6.2) c ( p −
1) def = (cid:16) δ ( p − , δ ( p − , δ ( p − , . . . , − δ ( p − p − , δ ( p − p , δ ( p − p + , . . . , δ ( p − n + (cid:17) , c ( p ) def = (cid:16) δ ( p )1 , δ ( p )2 , δ ( p )3 , . . . , δ ( p ) p − , − δ ( p ) p , δ ( p ) p + , . . . , δ ( p ) n + (cid:17) , where δ ( b ) ν > ≤ ν ≤ n + ≤ b ≤ p .Now we define the ( n + T = ( T , . . . , T n + ) by setting T ν def = Jrd ,γ (1) ν ⊕ · · · ⊕ Jrd ,γ ( p ν ⊕ Jrd , c (1) ν · · · ⊕ Jrd , c ( p ν , ≤ ν ≤ n + . (6.3)Recall that Γ = { γ ( b ) ν } and C = { c ( b ) ν } . The rest of this section is devoted to defining thematrices Γ and C appropriately so that the resulting tuple T defined by (6.3) is hypercyclic.We will henceforth just write T with the understanding that whenever Γ and C are givenmatrices, T is defined by (6.3). According to Lemma 4.7, the ( n + T is hypercyclic if and only if we have that { V ( m , Γ , C ) : m ∈ N n + } = R n . The following Lemma will help us simplify this statement:6.4. Lemma.
Let the matrices Γ and C be defined by (6.1) and (6.2) respectively. Suppose that theset { V (2 m , Γ , C : m ∈ N n + } is dense in ( R + × R ) p × ( R + ) p , where m def = (2 m , . . . , m n + ) . Then { V ( m , Γ , C ) : m ∈ N n + } is dense in R p + p = R n .Proof. Let x = ( x , . . . , x n ) ∈ R n be given. We need to approximate x with vectors of the form V ( m , Γ , C ) for a suitable sequence m = ( m , . . . , m n + ) ∈ N n + . Without loss of generality wecan assume that x j , ≤ j ≤ n . We define the vector σ = ( σ , . . . σ n + ) ∈ N n + as σ ν def = − sgn( x ν )2 , if 1 ≤ ν ≤ n , , if ν = n + . We claim that there exists a sequence m = m ( τ ) such that V (2 m ( τ ) + σ, Γ , C ) → x , as τ → ∞ .Indeed we have that V (2 m + σ, Γ , C ) = (cid:18) ( γ (1) ) m ( γ (1) ) σ , n + X ν = m ν γ (1) ν + n + X ν = σ ν γ (1) ν ,... ( γ ( p ) ) m ( γ ( p ) ) σ , n + X ν = m ν γ ( p ) ν + n + X ν = σ ν γ ( p ) ν , ( c (1) ) m ( c (1) ) σ , . . . , ( c ( p ) ) m ( c ( p ) ) σ (cid:19) . Consider now the vector y ∈ R n defined as y def = (cid:16) x ( γ (1) ) σ , x − n + X ν = σ ν γ (1) ν , . . . , x p − ( γ ( p ) ) σ , x p − n + X ν = σ ν γ ( p ) ν , x p + ( c (1) ) σ , . . . , x n ( c ( p ) ) σ (cid:17) . Setting V = (cid:16) , n + X ν = σ ν γ (1) ν , . . . , , n + X ν = σ ν γ ( p ) ν , , , . . . , (cid:17) and V = (cid:16) ( γ (1) ) σ , , . . . , ( γ ( p ) ) σ , , ( c (1) ) σ , . . . , ( c ( p ) ) σ (cid:17) , we see that V (2 m + σ, Γ , C ) = V ◦ V (2 m , Γ , C ) + V and x = V ◦ y + V . (6.5) YNAMICS OF TUPLES OF MATRICES IN JORDAN FORM 15
Here we denote by u ◦ v the Hadamard product of u , v ∈ R n : If u = ( u , . . . , u n ), v = ( v , . . . , v n )then u ◦ v = ( u v , u v , . . . , u n v n ) ∈ R n . For any 1 ≤ j ≤ p we have that y j − = x j ( γ ( j )1 ) σ · · · ( γ ( j ) n + ) σ n + = x j ( a ( j )1 ) σ · · · ( a ( j ) j − ) σ j − ( − a ( j ) j ) σ j ( a ( j + j + ) σ j + · · · ( − a ( j ) n + ) σ n + = ( − σ j x j ( a ( j )1 ) σ · · · ( a ( j ) n + ) σ n + = | x j | ( a ( j )1 ) σ · · · ( a ( j ) n + ) σ n + > . Similarly we can see that y j > p + ≤ j ≤ n . This shows that y ∈ ( R + × R ) p × ( R + ) p and thus there is a sequence m ( τ ) such that V (2 m ( τ ) , Γ , C ) → y as τ → ∞ . By (6.5) we havethat lim τ → + ∞ V (2 m ( τ ) + σ, Γ , C ) = lim τ → + ∞ V ◦ V (2 m ( τ ) , Γ , C ) + V = V ◦ y + V = x . Since x ∈ R n was arbitrary this concludes the proof of the lemma. (cid:3) Lemma 6.4 implies that in order to show that the ( n + T is hypercyclic it isenough to show that { V (2 m , Γ , C ) : m ∈ N n + } is dense in ( R + × R ) p × ( R + ) p , where Γ and C are defined by equations (6.1) and (6.2) respectively. We can reformulate this to get alinear condition in m ∈ N n + , like in Corollary 4.10. Indeed, observe that the matrix L nowbecomes L + = log a (1)1 log a (1)2 log a (1)3 · · · log a (1) p − log a (1) p log a (1) p + · · · log a (1) n + − / a (1)1 / a (1)2 / a (1)3 · · · / a (1) p − / a (1) p / a (1) p + · · · − / a (1) n + log a (2)1 log a (2)2 log a (2)3 · · · log a (2) p − log a (2) p log a (2) p + · · · log a (2) n + / a (2)1 − / a (2)2 / a (2)3 · · · / a (2) p − / a (2) p / a (2) p + · · · − / a (2) n + ... ... ... . . . ... ... ... . . . ... log a ( p )1 log a ( p )2 log a ( p )3 · · · log a ( p ) p − log a ( p ) p log a ( p ) p + · · · log a ( p ) n + / a ( p )1 / a ( p )2 / a ( p )3 · · · / a ( p ) p − − / a ( p ) p / a ( p ) p + · · · − / a ( p ) n + log δ (1)1 log δ (1)2 log δ (1)3 · · · log δ (1) p − log δ (1) p log δ (1) p + · · · log δ (1) n + ... ... ... . . . ... ... ... . . . ... log δ ( p )1 log δ ( p )2 log δ ( p )3 · · · log δ ( p ) p − log δ ( p ) p log δ ( p ) p + · · · log δ ( p ) n + , where a ( b ) and δ ( b ) have all their entries positive. We then have the desired inverse ofCorollary 4.10: Proposition.
Suppose that the set { L + m T : m ∈ N n + } is dense in R n . Then T is hypercyclic.Proof. Indeed, assuming that the set { L + m T : m ∈ N n + } is dense in R n we immediatelyconclude that the set { V (2 m , Γ , C ) : m ∈ N n + } is dense in ( R + × R ) p × ( R + ) p . To see thisnote that, for any 1 ≤ b ≤ p , the set n n + X ν = m ν log a ( b ) ν , ( m , . . . , m n + ) ∈ N n + o , is dense in R if and only if the set n ( a ( b )1 ) m · · · ( a ( b ) n + ) m n + , ( m , . . . , m n + ) ∈ N n + o , is dense in R + . However this is the same as saying that the set n ( γ ( b )1 ) m · · · ( γ ( b ) n + ) m n + , ( m , . . . , m n + ) ∈ N n + o , is dense in R + since | γ ( b ) ν | = a ( b ) ν > ν and b . We reason similarly for the c ( b ) ν ’s for 1 ≤ b ≤ p . However, by Lemma 6.4 this implies that the set { V ( m , Γ , C ) : m ∈ N n + } is dense in R n . By Lemma 4.7 we then get that T is hypercyclic. (cid:3) We will now construct the matrix L + so that { L + m T : m ∈ N n + } = R n . To that end it willbe helpful to consider the n + u , . . . , u n + ∈ R n which are just the correspondingcolumns of the n × ( n +
1) matrix L + . That is we have: u = (cid:18) log a (1)1 , − a (1)1 , log a (2)1 , a (2)1 , log a (3)1 , a (3)1 , . . . , log a ( p )1 , a ( p )1 , log δ (1)1 , . . . , log δ ( p )1 (cid:19) , u = (cid:18) log a (1)2 , a (1)2 , log a (2)2 , − a (2)2 , log a (3)2 , a (3)2 , . . . , log a ( p )2 , a ( p )2 , log δ (1)2 , . . . , log δ ( p )2 (cid:19) ,... (6.7) u p def = (cid:18) log a (1) p , a (1) p , log a (2) p , a (2) p , log a (3) p , a (3) p , . . . , log a ( p ) p , − a ( p ) p , log δ (1) p , . . . , log δ ( p ) p (cid:19) ,... u n + = (cid:18) log a (1) n + , − a (1) n + , log a (2)2 , − a (2) n + , . . . , log a ( p ) n + , − a ( p ) n + , log δ (1) n + , . . . , log δ ( p ) n + (cid:19) . The heart of the proof is the following theorem:
YNAMICS OF TUPLES OF MATRICES IN JORDAN FORM 17
Theorem.
For a positive integer n ≥ , let p ≥ and p ≥ be non-negative integers suchthat n = p + p . Then there exist real numbers a ( b ) ν > , ≤ b ≤ p , ≤ ν ≤ n + and δ ( b ) ν > , ≤ b ≤ p , ≤ ν ≤ n + , such that, if we define the vectors u , u , . . . , u n + by (6.7) , the followingconditions are satisfied:(i) The vectors u , . . . , u n are linearly independent over R .(ii) There exist positive irrational numbers c , . . . , c n such that u n + = − P n ν = c ν u ν .We conclude that(iii) Any n of the vectors u , . . . , u n , u n + are R -linearly independent.(iv) The vectors u , . . . , u n , u n + are Q -linearly independent. Before giving the proof, let us see how we can use Theorem 6.8 in order to prove Theorem1.10.
Proof of Theorem 1.10.
Because of Proposition 6.6, the proof of Theorem 1.10 reduces toshowing that the set { L + m T : m ∈ N n + } is dense in R n + . By the definition of the vectors u , . . . , u n + this is equivalent to showing that the set { L + m T : m ∈ N n + } = { m u + · · · + m n + u n + : ( m , . . . , m n + ) ∈ N n + } , is dense in R n . That is, we need to show that any x ∈ R n can be approximated by linearcombinations of the vectors u , . . . , u n + with coe ffi cients in N . To that end, we fix a x ∈ R n and ǫ >
0. We write x in the form x = R u + · · · R n u n + r u · · · + r n u n , where R , . . . , R n ∈ Z and r , . . . , r n ∈ [0 , N u n + isdense in R n / ( Z u + · · · + Z u n ). So, we can find arbitrarily large ℓ ∈ N such that ℓ u n + = R ′ u + · · · + R ′ n u n + r ′ u + · · · + r ′ n u n , where R ′ ν ∈ Z and r ′ ν ∈ (0 ,
1) for all 1 ≤ ν ≤ n , and | r ν − r ′ ν | < ǫ P n ν = k u ν k , for all 1 ≤ ν ≤ n . Now condition (ii) of Theorem 6.8 implies that R ′ ν < ≤ ν ≤ n . In fact we can makethe coe ffi cients R ′ ν as negative as we please by taking larger values of ℓ . Let us now write x ′ def = ℓ u n + + ( R − R ′ ) u + · · · + ( R n − R ′ n ) u n , where we make sure that the coe ffi cients R ν − R ′ ν > ≤ ν ≤ n by taking ℓ ∈ N aslarge as necessary. We then have k x − x ′ k = (cid:13)(cid:13)(cid:13)(cid:13) n X ν = ( r ν − r ′ ν ) u ν (cid:13)(cid:13)(cid:13)(cid:13) ≤ n X ν = | r ν − r ′ ν | k u ν k < ǫ. Since x ′ is a linear combination of u , . . . , u n + with coe ffi cients in N we are done. (cid:3) In order to organize the proof of Theorem 6.8 we need two additional technical lemmas.6.9.
Lemma.
Let δ , δ > . For any c > the non-linear equationx c + − δ x − δ c = , has a unique positive solution x = x ( c ) . We have that lim c → + ∞ x ( c ) = .Proof. First observe that the function f ( x ) = x c + − δ x − δ c is continuously di ff erentiablein x ∈ R + and satisfies f ( δ c ) = − δ c < f ( x ) > x large. Thus there is at least one x o ∈ R + , x o > ( δ ) c such that f ( x o ) =
0. Looking at the derivative of f , f ′ ( x ) = ( c + x c − δ we see that f has exactly one critical point at x = (cid:16) δ c + (cid:17) c < x o . The function f is negative for0 < x ≤ x and strictly increasing for x > x thus the solution x o is unique. We can definethen the function x ( c ) to be this unique solution.In order to prove that the function x = x ( c ) has a limit as c → + ∞ we argue as follows.First observe that for any c > f (( δ c ) c + ) = − δ ( δ c ) c + <
0. On the otherhand, for any A > δ we have that f (( Ac ) c ) = ( Ac ) c ( A − δ ( Ac ) c ) c − δ ( Ac ) c → + ∞ as c → + ∞ .We thus see that for c large enough, we have that ( δ c ) c + < x ( c ) < ( Ac ) c . Letting c → + ∞ we conclude that lim c → + ∞ x ( c ) = (cid:3) In the following Lemma we give the basic construction which is ‘half-way there’ to getthe vectors u , . . . , u n + we need in Theorem 6.8. For this, the following notation will beuseful. For any positive integer n ≥ p ≥ p ≥ YNAMICS OF TUPLES OF MATRICES IN JORDAN FORM 19 n = p + p , we consider the matrix ˜ L ( p , p ) as follows˜ L ( p , p ) = log a (1)1 − a (1)1 log a (2)1 1 a (2)1 · · · log a ( p )1 1 a ( p log δ (1)1 · · · log δ ( p )1 log a (1)2 1 a (1)2 log a (2)2 − a (2)2 · · · log a ( p )2 1 a ( p log δ (1)2 · · · log δ ( p )2 ... ... ... ... . . . ... ... ... . . . ... log a (1) p a (1) p log a (2) p a (2) p · · · log a ( p ) p − a ( p p log δ (1) p · · · log δ ( p ) p log a (1) p + a (1) p + log a (2) p + a (2) p + · · · log a ( p ) p + a ( p p + log δ (1) p + · · · log δ ( p ) p + ... ... ... ... . . . ... ... ... . . . ... log a (1) n − a (1) n − log a (2) n − a (2) n − · · · log a ( p ) n − a ( p n − log δ (1) n − · · · log δ ( p ) n − log a (1) n a (1) n log a (2) n a (2) n · · · log a ( p ) n a ( p n log δ (1) n · · · log δ ( p ) n where a ( b ) ν > ≤ ν ≤ n , 1 ≤ b ≤ p , and similarly δ ( b ) ν > ≤ ν ≤ n and 1 ≤ b ≤ p .Observe that the value p = δ ( b ) ν terms. Wealso define the closely related matrix ˜ L o ( p , p ) which is the special case of ˜ L ( p , p ) if we set a ( b ) n = ≤ b ≤ p and δ ( b ) n = ≤ b ≤ p . That is we have˜ L o ( p , p ) def = log a (1)1 − a (1)1 log a (2)1 1 a (2)1 · · · log a ( p )1 1 a ( p log δ (1)1 · · · log δ ( p )1 log a (1)2 1 a (1)2 log a (2)2 − a (2)2 · · · log a ( p )2 1 a ( p log δ (1)2 · · · log δ ( p )2 ... ... ... ... . . . ... ... ... . . . ... log a (1) p a (1) p log a (2) p a (2) p · · · log a ( p ) p − a ( p p log δ (1) p · · · log δ ( p ) p log a (1) p + a (1) p + log a (2) p + a (2) p + · · · log a ( p ) p + a ( p p + log δ (1) p + · · · log δ ( p ) p + ... ... ... ... . . . ... ... ... . . . ... log a (1) n − a (1) n − log a (2) n − a (2) n − · · · log a ( p ) n − a ( p n − log δ (1) n − · · · log δ ( p ) n − · · · · · · With this notation, we have the following Lemma:6.10.
Lemma.
Let n ≥ be a positive integer and p ≥ and p ≥ be non-negative integerssuch that n = p + p . Then there exist a ( b ) ν > for ≤ ν ≤ n − , ≤ b ≤ p , and δ ( b ) ν > for ≤ ν ≤ n − , ≤ b ≤ p (if p , ), such that det(˜ L o ( p , p )) , .Proof. We will prove the Lemma by induction on n . For each such n we have to consider allpossible combinations of p ≥ p ≥ n = p + p , and this will be reflectedin the inductive hypothesis. The first step of the induction is obvious. Indeed, for n = we necessarily have that p = p =
0. Then we need to show that there exists a choiceof a (1)1 such that the matrix˜ L o (1 , = log a (1)1 − / a (1)1 , ! , has non-zero determinant. However this is the case for any a (1)1 , . Now assume the conclusion of the Lemma is true for n −
1. There are two cases we needto consider depending on whether p = p ≥
1. First we consider the case n = p + p where p , p ≥
1. Since 2 p + ( p − = n − p − ≥
0, we can use the inductivehypothesis to get a matrix ˜ L o ( p , p −
1) with non-zero determinant. We need to construct˜ L o ( p , p ). Let a ( b ) ν , 1 ≤ ν ≤ n −
2, 1 ≤ b ≤ p and δ ( b ) ν , 1 ≤ ν ≤ n −
2, 1 ≤ b ≤ p −
1, be definedas the corresponding entries of the matrix ˜ L o ( p , p − δ ( p ) ν for all 1 ≤ ν ≤ n − a ( b ) n for all 1 ≤ b ≤ p and to δ ( b ) n for 1 ≤ b ≤ p − L o ( p , p ) = log a (1)1 − a (1)1 · · · log a ( p )1 1 a ( p log δ (1)1 · · · log δ ( p − log ∗ log a (1)2 1 a (1)2 · · · log a ( p )2 1 a ( p log δ (1)2 · · · log δ ( p − log ∗ ... ... . . . ... ... ... . . . ... ... log a (1) p a (1) p · · · log a ( p ) p − a ( p p log δ (1) p · · · log δ ( p − p log ∗ log a (1) p + a (1) p + · · · log a ( p ) p + a ( p p + log δ (1) p + · · · log δ ( p − p + log ∗ ... ... . . . ... ... ... . . . ... ... log a (1) n − a (1) n − · · · log a ( p ) n − a ( p n − log δ (1) n − · · · log δ ( p − n − log ∗ log ∗ / ∗ · · · log ∗ / ∗ log ∗ · · · log ∗ log δ ( p ) n − · · · · · · where the wildcards ‘*’ denote arbitrary positive choices. Now all the entries of ˜ L o ( p , p )are defined except for log δ ( p ) n − which we will now choose as follows. Developing thedeterminant of ˜ L o ( p , p ) with respect to the elements of the ( n − L o ( p , p ) = A + log δ ( p ) n − det(˜ L o ( p , p , − , (6.11)where A is a constant that does not depend on δ ( p ) n − but depends on all the other entriesof the matrix which we have already fixed. Since det(˜ L o ( p , p − , δ ( p ) n − such that the right hand side of (6.11) is non-zeroso we are done in this case. YNAMICS OF TUPLES OF MATRICES IN JORDAN FORM 21
Turning to the case p =
0, that is n = p , we need to construct the matrix ˜ L o ( p ,
0) withthe desired structure and non-zero determinant. Here the inductive hypothesis impliesthe existence of a matrix ˜ L o ( p − ,
1) with non zero determinant:˜ L o ( p − , = log a (1)1 − a (1)1 · · · log a ( p − a ( p − log δ (1)1 log a (1)2 1 a (1)2 · · · log a ( p − a ( p − log δ (1)2 ... ... . . . ... ... ... log a (1) p − a (1) p − · · · log a ( p − p − − a ( p − p − log δ (1) p − log a (1) p a (1) p · · · log a ( p − p a ( p − p log δ (1) p log a (1) p + a (1) p + · · · log a ( p − p + a ( p − p + log δ (1) p + ... ... . . . ... ... ... log a (1) n − a (1) n − · · · log a ( p − n − a ( p − n − log δ (1) n − · · · We define a ( b ) ν as the corresponding entries of ˜ L o ( p − ,
1) for 1 ≤ ν ≤ n − ≤ b ≤ p − a ( p ) ν = δ (1) ν for 1 ≤ ν ≤ n −
2. Finally we give arbitrary positive values to a ( b ) n for all 1 ≤ b ≤ p −
1. This defines all entries of ˜ L o ( p − ,
1) except the two rightmost entrieson the ( n − a ( p ) n − which we will choosenow. The matrix ˜ L o ( p ,
0) has the following structure:˜ L o ( p , = log a (1)1 − a (1)1 · · · log a ( p − a ( p − log δ (1)1 1 δ (1)1 log a (1)2 1 a (1)2 · · · log a ( p − a ( p − log δ (1)2 1 δ (1)2 ... ... . . . ... ... ... ... log a (1) p − a (1) p − · · · log a ( p − p − − a ( p − p − log δ (1) p − δ (1) p − log a (1) p a (1) p · · · log a ( p − p a ( p − p log δ (1) p − δ (1) p log a (1) p + a (1) p + · · · log a ( p − p + a ( p − p + log δ (1) p + δ (1) p + ... ... . . . ... ... ... ... log a (1) n − a (1) n − · · · log a ( p − n − a ( p − n − log δ (1) n − δ (1) n − log ∗ / ∗ · · · log ∗ / ∗ log a ( p ) n − / a ( p ) n − · · · where again the wildcards ‘*’ denote arbitrary but fixed choices of positive real numbers.We develop the determinant of ˜ L o ( p ,
0) with respect to the elements of the ( n − We easily see that we havedet(˜ L o ( p , = A + B log a ( p ) n − + / a ( p ) n − det(˜ L o ( p − , , (6.12)where the constants A , B are fixed real numbers that depend on all the entries of ˜ L o ( p , a ( p ) n − . Again, since det(˜ L o ( p − , , A , B is, there is always a choice of a ( p ) n − > (cid:3) Remark.
We showed in Lemma 6.10 that there is a choice of a ( b ) ν , 1 ≤ ν ≤ n −
1, 1 ≤ b ≤ p ,and δ ( b ) ν for 1 ≤ ν ≤ n −
1, 1 ≤ b ≤ p , such that det(˜ L o ( p , p )) is non-zero. However, ifwe consider det(˜ L o ( p , p )) as a function of the entries a ( b ) ν , δ ( b ) ν , we see this is a real-analyticfunction in ( R + ) ( p + p )( n − . We conclude that the set { det(˜ L o ( p , p )) = } is a closed set whichhas zero ( p + p )( n − n and any choice of p ≥ p ≥ n = p + p ,there are ‘generic’ choices of a ( b ) ν , δ ( b ) ν such that the matrix ˜ L o ( p , p ) is invertible. Proof of Theorem 6.8.
We will prove the Theorem based on the construction of Lemma 6.10.The conclusions (iii) and (iv) of the Theorem are easy consequences of (i) and (ii) so wewill accept them with no further comment. Let n and p ≥ p ≥
0, be given non-negativeintegers such that n = p + p . We need to define the vectors u , u , . . . , u n , u n + ∈ R n ofthe form (6.7), with u , . . . , u n linearly independent over R , as well as positive irrationalconstants c , . . . , c n such that u n + = − n X ν = c ν u ν . (6.14)We will choose the vectors u , . . . , u n to be the rows of an appropriately constructed matrix˜ L ( p , p ). To that end, we consider the matrix ˜ L ( p , p ) whose first n − n − L o ( p , p ) provided by Lemma 6.10. We first givearbitrary positive irrational values to the constants c , . . . , c p . Then we choose the constants c p + , . . . , c n − in R + \ Q so that n − X ν = p + c ν a ( b ) ν − p X ν = c ν a ( b ) ν > , (6.15)for all 1 ≤ b ≤ p . Observe that this is always possible when p + p ≥ n − − ( p + + = p + p − ≥ n ≥
3. The value of c n we leave undetermined for now. YNAMICS OF TUPLES OF MATRICES IN JORDAN FORM 23
We define the auxiliary variables1˜ a ( b ) n + = n − X ν = ν , b c ν a ( b ) ν − c b a ( b ) b , ≤ b ≤ p , and log 1ˆ a ( b ) n + = n − X ν = c ν log a ( b ) ν . (6.16)Similarly, if p , δ ( b ) n + = n − X ν = c ν δ ( b ) ν , ≤ b ≤ p , and log 1ˆ δ ( b ) n + = n − X ν = c ν log δ ( b ) ν , ≤ b ≤ p . Observe that by definition ˆ a ( b ) n + > ≤ b ≤ p and ˜ δ ( b ) n + , ˆ δ ( b ) n + > ≤ b ≤ p .We also have that1˜ a ( b ) n + = n − X ν = ν , b c ν a ( b ) ν − c b a ( b ) b ≥ n − X ν = p + c ν a ( b ) ν − p X ν = c ν a ( b ) ν > , by (6.15).Now we define u n . For c n > ≤ b ≤ p , we define a ( b ) n = a ( b ) n ( c n ) to be theunique positive solution of the non-linear equation( a ( b ) n ) c n + − ˆ a ( b ) n + ˜ a ( b ) n + a ( b ) n − ˆ a ( b ) n + c n = , ≤ b ≤ p , (6.17)as a function of c n . Similarly, for each 1 ≤ b ≤ p we define δ ( b ) n = δ ( b ) n ( c n ) to be the uniquesolution of the non-linear equation( δ ( b ) n ) c n + − ˆ δ ( b )1 ˜ δ ( b ) n + δ ( b ) n − ˆ δ ( b ) n + c n = , ≤ b ≤ p , (6.18)again as a function of c n . In order to define these solutions, we rely on Lemma 6.9. Remark.
A few words are necessary to justify the definitions (6.17) and (6.18). Observe,for example, that we need to define the numbers c n and a ( b ) n so that condition (6.14) issatisfied. For any 1 ≤ b ≤ p , (6.14) reads:log 1 a ( b ) n + = n X ν = c ν log a ( b ) ν = n − X ν = c ν log a ( b ) ν + c n log a ( b ) n = log 1ˆ a ( b ) n + + c n log a ( b ) n (6.20) = log ( a ( b ) n ) c n ˆ a ( b ) n + , and 1 a ( b ) n + = n − X ν = ν , b c ν a ( b ) ν − c b a ( b ) b + c n a ( b ) n = a ( b ) n + + c n a ( b ) n . (6.21)Combining (6.20) and (6.21) we conclude that c n and a ( b ) b must satisfy (6.17). On the otherhand, for 1 ≤ b ≤ p , equation (6.14) giveslog 1 δ ( b ) n + = δ ( b ) n + + c n δ ( b ) n (6.22)Technically speaking, it is enough to take arbitrary values for δ ( b ) n and define δ ( b ) n + exactly by(6.22). However, we prefer to define δ ( b ) n + again by the more restrictive non-linear equation(6.18) for consistency.For each c n >
0, the matrix ˜ L ( p , p ) is defined in all its entries. We consider det(˜ L ( p , p ))as a function of c n . Suppose that we have det(˜ L ( p , p ))( c n ) = c n >
0. In this case there is a sequence of irrational c n → + ∞ for which we have thatlim c n → + ∞ det(˜ L ( p , p ))( c n ) =
0. However, according to Lemma 6.9, we have thatlim c n → + ∞ a ( b ) n ( c n ) = , ≤ b ≤ p , and lim c n → + ∞ δ ( b ) n ( c n ) = , ≤ b ≤ p . But this means that0 = lim c n → + ∞ det(˜ L ( p , p ))( c n ) = det(˜ L o ( p , p )) , which is a contradiction since we have chosen ˜ L o ( p , p ) so that its determinant is non-zero.We conclude that for large enough irrational and positive c n , the matrix ˜ L ( p , p ) has non-zero determinant. This means that its rows, the vectors u , . . . , u n , are linearly independentover R . YNAMICS OF TUPLES OF MATRICES IN JORDAN FORM 25
Finally, we define a ( b ) n + , 1 ≤ b ≤ p , exactly by the desired property (6.14):1 a ( b ) n + = n X ν = ν , b c ν a ( b ) ν − c b a ( b ) b = n − X ν = ν , b c ν a ( b ) ν − c b a ( b ) b + c n a ( b ) n = a ( b ) n + + c n a ( b ) n > . (6.23)Similarly, we define δ ( b ) n + , 1 ≤ b ≤ p , as1 δ ( b ) n + = n X ν = c ν δ ( b ) ν . (6.24)It remains to check the validity of (6.14). We have for every 1 ≤ b ≤ p , thatlog 1 a ( b ) n + = log (cid:16) a ( b ) n + + c n a ( b ) n (cid:17) = log (cid:16) ( a ( b ) n + ) c n + ˆ a ( b ) n + (cid:17) , where in the last equality we have used the definition of a ( b ) n + , equation (6.17). Now, usingthe definition of ˆ a ( b )1 in equation (6.16), we getlog a ( b ) n + = − c n + log a ( b ) n + + log ˆ a ( b ) n + = − c n + log a ( b ) n + − n + X ν = c ν log a ( b ) ν = − n + X ν = c ν log a ( b ) ν . A similar calculation for 1 ≤ b ≤ p shows that log δ ( b )1 = − P n + ν = c ν log δ ( b ) ν . This togetherwith definitions (6.23) and (6.24) shows that u n + = − P n ν = c ν u ν and completes the proof ofthe Theorem. (cid:3) The proof in the case n = . Here we have that p = p =
0. We define thevector γ (1) def = ( a (1)1 , a (1)2 , − a (1)3 ) , where a (1) ν > ≤ ν ≤
3. We define the triple of matrices J = ( J , J , J ) by means of J ν def = Jrd ,γ (1) ν = γ (1) ν γ (1) ν ! , ≤ ν ≤ . With
Γ = { γ (1) ν } , Lemma 4.7 says that J is hypercyclic if and only if { V ( m , Γ ) : m ∈ N } = R .Recall that for m ∈ N V ( m , Γ ) = (cid:16) ( γ (1) ) m , X ν = m j γ (1) j (cid:17) . We have the analogue of Lemma 6.4 whose proof we omit.6.25.
Lemma.
Suppose that the set { V (2 m , Γ ) : m ∈ N } is dense in R + × R . Then { V (2 m + , Γ ) : m ∈ N } is dense in R − × R . We conclude that the set { V ( m , Γ ) : m ∈ N } is dense in R . Consider the matrix L + , L + = log a (1)1 log a (1)2 log a (1)3 / a (1)1 / a (1)2 − / a (1)3 ! . Observe that in order to show that the set V (2 m , Γ ) is dense in R + × R , it is enough to showthat { L + m T : m ∈ N } is dense in R . We define the vectors u , u , u as the correspondingcolumns of L + , u = (cid:16) log a (1)1 , a (1)1 (cid:17) , u = (cid:16) log a (1)2 , a (1)2 (cid:17) , (6.26) u = (cid:16) log a (1)3 , − a (1)3 (cid:17) . We have the analogue of Theorem 6.8 which again is the main part of the proof.6.27.
Theorem.
There exist real numbers a (1) ν > , ≤ ν ≤ such that, if we define the vectorsu , u , u by (6.26) , the following conditions are satisfied:(i) The vectors u , u are linearly independent over R .(ii) There exist positive irrational numbers c , c such that u = − ( c u + c u ) .We conclude that(iii) Any of the vectors u , u , u are R -linearly independent.(iv) The vectors u , u , u are Q -linearly independent. Using theorem 6.27, we can conclude the proof of Theorem 1.10 for R just like in theproof of the case n ≥
3. We will only describe how one proves Theorem 6.8, giving anargument very close to the one given in the proof of Theorem 6.8.
Proof of Theorem 6.27.
Consider the matrix˜ L o (1 , = log a (1)1 / a (1)1 , ! , where a (1)1 ,
1. Obviously we then have det(˜ L o (1 , ,
0. We give an arbitrary positiveand irrational value to the constant c . Now for every c > a (1)2 = a (1)2 ( c ) as theunique solution of the non-linear equation( a (1)2 ) c + − c ( a (1)1 ) c + a (1)2 − a (1)1 ) c c = , (6.28) YNAMICS OF TUPLES OF MATRICES IN JORDAN FORM 27 as a function of c . This is possible because of Lemma 6.9. The same Lemma also givesthat lim c → + ∞ c ∈ R + \ Q a (1)2 ( c ) =
1. Therefore the matrix˜ L (1 , = log a (1)1 / a (1)1 log a (1)2 / a (1)2 ! , satisfies lim c → + ∞ c ∈ R + \ Q det(˜ L (1 , = det(˜ L o (1 , , . Choosing c large enough in R + \ Q we get that the vectors u , u are linearly independentover R . Now we define1 a (1)3 def = c a (1)1 + c a (1)2 . It remains to check the validity of u = − ( c u + c u ). For this observe thatlog 1 a (1)3 = log (cid:16) c a (1)1 + c a (1)2 (cid:17) = log (cid:16) ( a (1)1 ) c + ( a (1)2 ) c + a (1)1 a (1)2 (cid:17) = c log a (1)1 + c log a (1)2 , where the second equality is due to the non-linear equation (6.28). (cid:3) R eferences Bayart, F. and ´E. Matheron. 2009.
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