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Applied General Topology

c© Universidad Politécnica de Valencia

Volume 12, no. 1, 2011

pp. 35-39

Hypercyclic abelian semigroup of matrices
on Cn and Rn 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 Cn

or R
n

is never k-transitive for k ≥ 2. These answer questions raised by

Feldman and Javaheri.

2010 MSC: 37C85, 47A16

Keywords: Hypercyclic, tuple of matrices, semigroup, subgroup, dense or-
bit, transitive, semigroup action.

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 A1, A2, . . . , Ap

on Kn. We will let G = {Ak11 A
k2
2 . . . A

kp
p : k1, k2, . . . , kp ∈ N} be the semi-

group generated by A1, A2, . . . , Ap. For a vector x ∈ K
n, the orbit of x under

the action of G on Kn is OG(x) = {Ax : A ∈ G}. For a subset E ⊂ K
n, denote

by E (resp.
◦

E ) the closure (resp. interior) of E. A subset E ⊂ Kn is called
G-invariant if A(E) ⊂ E for any A ∈ G. The orbit OG(x) ⊂ K

n is dense (resp.

locally dense) in Kn if OG(x) = K
n (resp.

˚
OG(x) 6= ∅). The semigroup G is

called hypercyclic (or also topologically transitive) (resp. locally hypercyclic) if
there exists a vector x ∈ Kn such that OG(x) is dense (resp. locally dense) in

∗This work is supported by the research unit: systèmes dynamiques et combinatoire:
99UR15-15



36 A. Ayadi

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 (Kn)
k

the k-fold
Cartesian product of Kn. For every u = (x1, . . . , xk) ∈ (K

n)k, the orbit of u
under the action of G on (Kn)k is denoted

O
k
G(u) = {(Ax1, . . . , Axk) : A ∈ G}.

When k = 1, OkG(u) = OG(u). We say that the action of G on K
n is k-

transitive if, the induced action of G on (Kn)
k

is hypercyclic, this is equivalent

to that for some u ∈ (Kn)
k
, Ok

G
(u) = (Kn)k. A 2-transitive action is also called

weak topological mixing and 1-transitive means hypercyclic.
In [6], Feldman showed that in Cn there exist a hypercyclic semigroup gen-

erated by (n + 1)-tuple of diagonal matrices on Cn and that no semigroup
generated by n-tuple of diagonalizable matrices on Kn 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 diagonaliz-
able matrices on Rn. However, they show in [5] that there exist a hypercyclic
semigroup generated by (n + 1)-tuple of diagonalizable matrices A1, . . . , An+1
on Rn.

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 Cn required to form a hypercyclic tuple in Cn 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 Kn

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 Cn is not locally hypercyclic.

Theorem 1.2. Let G be an abelian semigroup generated by p matrices (p ≥ 1)
on Kn (K = R or C). Then the action of G on Kn is never k-transitive for
k ≥ 2.

2. On hyercyclic semigroups

Let Mn(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 Mn(K). Let G be an abelian
semigroup generated by p matrices (p ≥ 1) on Kn and we let G′ = G∩GL(n, K).



Hypercyclic abelian semigroup of matrices on Cn and Rn and k-transitivity (k ≥ 2) 37

Lemma 2.1. Under the notation above, let k ≥ 1 be an integer and u ∈ (Kn)k.
Then

(i) Ok
G

(u) = (Kn)k if and only if Ok
G′

(u) = (Kn)k.

(ii)
˚

Ok
G

(u) = ∅ if and only if
˚

Ok
G′

(u) = ∅.

In particular, if the action of G on Kn is k-transitive so is the action of G′ on
K

n.

Proof. (i) Suppose that Ok
G′

(u) = (Kn)
k

for some u ∈ (Kn)
k
. Then since

Ok
G′

(u) ⊂ Ok
G

(u), we see that Ok
G

(u) = (Kn)
k
.

Conversely, suppose there exists u ∈ (Kn)
k

such that Ok
G

(u) = (Kn)
k
. De-

note by (A1, . . . , Ap) an p-tuple of matrices on K
n which generate the semi-

group G. One can suppose that for some 0 ≤ r ≤ p, A1, . . . , Ar ∈ GL(n, K)
and Ar+1, . . . , Ap ∈ Mn(K)\GL(n, K). Then G

′ = G ∩ GL(n, K) is the semi-
group generated by A1, . . . , Ar. For k = 1, . . . , r, write Im(Ak) = Ak(K

n) the
range of Ak. Then Im(Ak) is a vector subspace of K

n of dimension < n, hence
◦

Im(Ak) = ∅.

- If r = p then G = G′ and so (i) is obvious.

- If r = 0 then for every u ∈ (Kn)k, OkG(u) ⊂
p
⋃

k=1

(Im(Ak))
k ∪ {u}. Since

˚p
⋃

k=1

(Im(Ak))k = ∅,
◦

Ok
G

(u) = ∅.

- If 0 < r < p then

O
k
G(u) ⊂





r
⋃

j=1

(Im(Aj ))
k



 ∪ OkG′ (u).

It follows that

(Kn)
k
⊂





r
⋃

j=1

(Im(Aj ))
k



 ∪ Ok
G′

(u)

and therefore Ok
G′

(u) = (Kn)
k
.

The proof of (ii) is the same as for (i). �

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 Cn. Then every locally dense orbit of G is dense in Cn.

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.



38 A. Ayadi

Proof of Theorem 1.1. Let G be an abelian semigroup generated by n matrices

on Cn and we let G′ = G ∩ GL(n, C). By Corollary 2.4,
˚

Ok
G′

(u) = ∅ for every

u ∈ (Cn)k and hence by Lemma 2.1,
˚

Ok
G

(u) = ∅. The proof is complete. �

3. On k-transitivity (k ≥ 2)

Let recall first the following result:

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 Kn such
that if, u, v ∈ U and (Bm)m∈N is a sequence of G such that lim

m→+∞
Bmu = v

then lim
m→+∞

B−1m 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 Kn as in Proposition 3.1. Then for

every k ≥ 2, if v ∈ U k and w ∈ Ok
G

(v) ∩ U k then Ok
G

(v) ∩ U k = Ok
G

(w) ∩ U k.

Proof. Write v = (v1, . . . , vk), w = (w1, . . . , wk) ∈ U
k. Suppose that w ∈

Ok
G

(v) ∩ U k. Then there exists a sequence (Bm)m∈N in G such that

lim
m→+∞

(Bmv1, . . . , Bmvk) = (w1, . . . , wk).

Then lim
m→+∞

Bmvj = wj , for every 1 ≤ j ≤ k. Since vj , wj ∈ U , so by

Proposition 3.1, lim
m→+∞

B−1m wj = vj and hence

lim
m→+∞

(B−1m w1, . . . , B
−1
m wk) = v ∈ O

k
G

(w).

It follows that Ok
G

(v) ∩ U k = Ok
G

(w) ∩ U k. �

Proof of Theorem 1.2. Suppose the action of G is k-transitive (k ≥ 2), then

there exists v = (v1, . . . , vk) ∈ (K
n)k so that Ok

G
(v) = (Kn)k. We let G′ = G ∩

GL(n, K). By Lemma 2.1, Ok
G′

(v) = (Kn)k. Denote by G′′ the group generated
by G′ and by U a G′′-invariant dense open subset in Kn as in Proposition 3.1.

Then Ok
G′′

(v) = (Kn)k and hence v ∈ U k. Write

w := (v1, . . . , v1). Then w ∈ U
k and by Corollary 3.2, Ok

G′′
(w) = (Kn)k

(since U k is dense in (Kn)k). It follows that OG′′ (v1) = K
n.

Let ϕ : Kn −→ (Kn)
k

be the homomorphism defined by

ϕ(x) = (x, . . . , x), x ∈ Kn.

Then OkG′′ (w) = ϕ(OG′′ (v1)) ⊂ ϕ(K
n). As ϕ(Kn) is a vector subspace of (Kn)

k

of dimension n < nk, OG′′ (w) cannot be dense in (K
n)k (since k ≥ 2), this is

a contradiction and the theorem is proved. �



Hypercyclic abelian semigroup of matrices on Cn and Rn and k-transitivity (k ≥ 2) 39

References

1. A. Ayadi and H. Marzougui, Dynamic of Abelian subgroups of GL(n, C): a structure
Theorem, Geom. Dedicata 116 (2005), 111–127.

2. A. Ayadi and H. Marzougui, Dense orbits for abelian subgroups of GL(n, C), Foliations
2005: World Scientific, Hackensack, NJ (2006), 47–69.

3. F. Bayart and E. Matheron, Dynamics of Linear Operators, Cambridge Tracts in Math.,
179, Cambridge University Press, 2009.

4. G. Costakis, D. Hadjiloucas and A. Manoussos, Dynamics of tuples of matrices, Proc.
Amer. Math. Soc. 137, no. 3 (2009), 1025–1034.

5. 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), 229–
237.

6. N. S. Feldman, Hypercyclic tuples of operators and somewhere dense orbits, J. Math.
Anal. Appl. 346 (2008), 82–98.

7. M. Javaheri, Topologically transitive semigroup actions of real linear fractional transfor-
mations, J. Math. Anal. Appl. 368 (2010), 587–603.

(Received June 2010 – Accepted January 2011)

Adlene Ayadi (adleneso@yahoo.fr)
University of Gafsa, Department of Mathematics, Faculty of Science of Gafsa,
2112, Gafsa, Tunisia.


	Hypercyclic abelian semigroup of matrices[3pt] on Cn and Rn and k-transitivity (k2). By A. Ayadi