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1 HYPERCYCLIC ABELIAN SEMIGROUP OF MATRICES ON C n AND R n AND k-transitivity (k 2) ADLENE AYADI Abstract. We prove that the minimal number of matrices on C n required to form a hypercyclic abelian semigroup on C n is n + 1. We also prove that the action of any abelian semigroup finitely generated by matrices on C n or R n is never k-transitive for k 2. These answer questions raised by Feldman and Javaheri. 1. Introduction Let K = R or C. Following Feldman from [6], by an p-tuple of matrices, we mean a finite sequence of length p (p 1) of commuting matrices A 1, A 2,..., A p on K n. We will let G = {A k 1 1 Ak Akp p : k 1, k 2,..., k p N} be the semi-group generated by A 1, A 2,..., A p. For a vector x K n, the orbit of x under the action of G on K n is O G (x) = {Ax : A G}. For a subset E K n, denote by E (resp. E ) the closure (resp. interior) of E. A subset E K n is called G-invariant if A(E) E for any A G. The orbit O G (x) K n is dense (resp. locally dense) in K n if O G (x) = K n (resp. O G (x) ). The semigroup G is called hypercyclic (or also topologically transitive) (resp. locally hypercyclic) if there exists a vector x K n such that O G (x) is dense (resp. locally dense) in K n. For an account of results and bibliography on hypercyclicity, we refer to the book [3] by Bayart and Matheron. On the other part, let k 1 be an integer. Denote by (K n ) k the k-fold Cartesian product of K n. For every u = (x 1,..., x k ) (K n ) k, the orbit of u under the action of G on (K n ) k is denoted O k G(u) = {(Ax 1,..., Ax k ) : A G} When k = 1, O k G (u) = O G(u). We say that the action of G on K n is k- transitive if, the induced action of G on (K n ) k is hypercyclic, this is equivalent to that for some u (K n ) k, O k G (u) = (Kn ) k. A 2-transitive action is also called weak topological mixing and 1-transitive means hypercyclic Mathematics Subject Classification. 37C85, 47A16. Key words and phrases. Hypercyclic, tuple of matrices, semigroup, subgroup, dense orbit, transitive, semigroup action. This work is supported by the research unit: systèmes dynamiques et combinatoire: 99UR

2 2 ADLENE AYADI In [6], Feldman showed that in C n there exist a hypercyclic semigroup generated by (n+1)-tuple of diagonal matrices on C n and that no semigroup generated by n-tuple of diagonalizable matrices on K n can be hypercyclic. If one remove the diagonalizability condition, Costakis et al. proved in [4] that there exists a hypercyclic semigroup generated by n-tuple of non diagonalizable matrices on R n. However, they show in [5] that there exist a hypercyclic semigroup generated by (n + 1)-tuple of diagonalizable matrices A 1,..., A n+1 on R n. The main purpose of this paper is twofold: firstly, we give a general result (with respect to the results above) by showing that the minimal number of matrices on C n required to form a hypercyclic tuple in C n is n + 1. This answer a question raised by Feldman in ([6], Section 6). Secondly, we prove that the action of any abelian semigroup finitely generated by matrices on K n is never k-transitive for k 2. This answer a question of Javaheri in ([7], Problem 3). Our principal results are the following: Theorem 1.1. For every n 1, any abelian semigroup generated by n matrices on C n is not locally hypercyclic. Theorem 1.2. Let G be an abelian semigroup generated by p matrices (p 1) on K n (K = R or C). Then the action of G on K n is never k-transitive for k On hyercyclic semigroups Let M n (K) be the set of all square matrices of order n 1 with entries in K and GL(n, K) be the group of invertible matrices of M n (K). Let G be an abelian semigroup generated by p matrices (p 1) on K n and we let G = G GL(n, K). Lemma 2.1. Under the notation above, let k 1 be an integer and u (K n ) k. Then (i) OG k (u) = (Kn ) k if and only if OG k (u) = (K n ) k. (ii) OG k (u) = if and only if OG k (u) =. In particular, if the action of G on K n is k-transitive so is the action of G on K n. Proof. (i) Suppose that O k G (u) = (K n ) k for some u (K n ) k. Then since O k G (u) O k G (u), we see that Ok G (u) = (Kn ) k.

3 HYPERCYCLIC ABELIAN SEMIGROUP OF MATRICESON C n AND R n AND k-transitivity (k 2)3 Conversely, suppose there exists u (K n ) k such that OG k (u) = (Kn ) k. Denote by (A 1,..., A p ) an p-tuple of matrices on K n which generate the semigroup G. One can suppose that for some 0 r p, A 1,..., A r GL(n, K) and A r+1,..., A p M n (K)\GL(n, K). Then G = G GL(n, K) is the semigroup generated by A 1,..., A r. For k = 1,..., r, write Im(A k ) = A k (K n ) the range of A k. Then Im(A k ) is a vector subspace of K n of dimension < n, hence Im(A k ) =. - If r = p then G = G and so (i) is obvious. - If r = 0 then for every u (K n ) k, O k G (u) p p (Im(A k )) k =, k=1 OG k (u) =. k=1 - If 0 < r < p then r OG(u) k (Im(A j )) k OG k (u). It follows that and therefore O k G (u) = (K n ) k. j=1 r (K n ) k (Im(A j )) k OG k (u) j=1 The proof of (ii) is the same as for (i). (Im(A k )) k {u}. Since Lemma 2.2. ([2], Corollary 1.5). Let G be an abelian subgroup of GL(n, C). If G is generated by n matrices (n 1), it has no dense orbit. Lemma 2.3. ([6], Corollary 5.7). Let G be an abelian semigroup generated by p matrices (p 1) on C n. Then every locally dense orbit of G is dense in C n. From Lemmas 2.2 and 2.3, we obtain the following: Corollary 2.4. Any abelian semigroup generated by n matrices (n 1) of GL(n, C) is not locally hypercyclic. Proof of Theorem 1.1. Let G be an abelian semigroup generated by n matrices on C n and we let G = G GL(n, C). By Corollary 2.4, OG k (u) = for every u (C n ) k and hence by Lemma 2.1, OG k (u) =. The proof is complete.

4 4 ADLENE AYADI Let recall first the following result: 3. On k-transitivity (k 2) Proposition 3.1 ([1], Theorem 4.1). Let G be an abelian subgroup of GL(n, K) (K = R or C). Then there exists a G-invariant dense open subset U in K n such that if, u, v U and (B m ) m N is a sequence of G such that lim B mu = v then lim m + m + B 1 m v = u. Corollary 3.2. Let G be an abelian subgroup of GL(n, K) (K = R or C) and let U be a G-invariant dense open subset of K n as in Proposition 3.1. Then for every k 2, if v U k and w O k G (v) U k then O k G (v) U k = O k G (w) U k. Proof. Write v = (v 1,..., v k ), w = (w 1,..., w k ) U k. Suppose that w O k G (v) U k. Then there exists a sequence (B m ) m N in G such that lim (B mv 1,..., B m v k ) = (w 1,..., w k ). m + Then lim B mv j = w j, for every 1 j k. m + Proposition 3.1, m w j = v j and hence lim m + B 1 lim m + (B 1 m w 1,..., B 1 m w k ) = v O k G (w). It follows that O k G (v) U k = O k G (w) U k. Since v j, w j U, so by Proof of Theorem 1.2. Suppose the action of G is k-transitive (k 2), then there exists v = (v 1,..., v k ) (K n ) k so that O k G (v) = (Kn ) k. We let G = G GL(n, K). By Lemma 2.1, O k G (v) = (K n ) k. Denote by G the group generated by G and by U a G -invariant dense open subset in K n as in Proposition 3.1. Then O k G (v) = (K n ) k and hence v U k. Write w := (v 1,..., v 1 ). Then w U k and by Corollary 3.2, O k G (w) = (K n ) k (since U k is dense in (K n ) k ). It follows that O G (v 1 ) = K n. Let ϕ : K n (K n ) k be the homomorphism defined by ϕ(x) = (x,..., x), x K n. Then O k G (w) = ϕ(o G (v 1 )) ϕ(k n ). As ϕ(k n ) is a vector subspace of (K n ) k of dimension n < nk, O G (w) cannot be dense in (K n ) k (since k 2), this is a contradiction and the theorem is proved.

5 HYPERCYCLIC ABELIAN SEMIGROUP OF MATRICESON C n AND R n AND k-transitivity (k 2)5 References 1. A. Ayadi, H. Marzougui, Dynamic of Abelian subgroups of GL(n, C): a structure Theorem, Geometria Dedicata. 116 (2005), A. Ayadi, H. Marzougui, Dense orbits for abelian subgroups of GL(n, C), Foliations 2005: World Scientific, Hackensack, NJ (2006), F. Bayart, E. Matheron, Dynamics of Linear Operators, Cambridge Tracts in Math., 179, Cambridge University Press, G. Costakis, D. Hadjiloucas, and A. Manoussos, Dynamics of tuples of matrices, Proc. Amer. Math. Soc. 137 no 3, (2009) G. Costakis, D. Hadjiloucas and A. Manoussos, On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple, J. Math. Anal. Appl. 365 (2010) N.S. Feldman, Hypercyclic tuples of operators and somewhere dense orbits, J. Math. Anal. Appl. 346 (2008), M. Javaheri, Topologically transitive semigroup actions of real linear fractional transformations, J. Math. Anal. Appl. (2010), to appear. Adlene Ayadi, Department of Mathematics, Faculty of Science of Gafsa, Gafsa, Tunisia. address: adleneso@yahoo.fr, hmarzoug@ictp.it