@
Appl. Gen. Topol. 21, no. 1 (2020), 17-34

doi:10.4995/agt.2020.11807

c© AGT, UPV, 2020

Dynamic properties of the dynamical system
(SFnm(X),SF

n
m(f))

Franco Barragán, Alicia Santiago-Santos∗ and Jesús F. Tenorio†

Instituto de F́ısica y Matemáticas, Universidad Tecnológica de la Mixteca, Carretera a Acatlima,

Km 2.5, Huajuapan de León, Oaxaca, México. (franco@mixteco.utm.mx,alicia@mixteco.utm.mx,

jtenorio@mixteco.utm.mx)

Communicated by F. Balibrea

Abstract

Let X be a continuum and let n be a positive integer. We consider
the hyperspaces Fn(X) and SFn(X). If m is an integer such that
n > m ≥ 1, we consider the quotient space SFnm(X). For a given map
f : X → X, we consider the induced maps Fn(f) : Fn(X) → Fn(X),
SFn(f) : SFn(X) → SFn(X) and SFnm(f) : SFnm(X) → SFnm(X).
In this paper, we introduce the dynamical system (SFnm(X),SFnm(f))
and we investigate some relationships between the dynamical systems
(X, f), (Fn(X),Fn(f)), (SFn(X),SFn(f)) and (SFnm(X),SFnm(f))
when these systems are: exact, mixing, weakly mixing, transitive, to-
tally transitive, strongly transitive, chaotic, irreducible, feebly open and
turbulent.

2010 MSC: 54B20; 37B45; 54F50; 54F15.

Keywords: chaotic; continuum; dynamical system; exact; feebly open; hy-
perspace; induced map; irreducible; mixing; strongly transitive;
symmetric product; symmetric product suspension; totally tran-
sitive; transitive; turbulent; weakly mixing.

∗This paper was partially supported by the project: “Propiedades dinámicas y topológicas
sobre sistemas dinámicos inducidos”, (UTMIX-PTC-064) of PRODEP, 2017.
†Corresponding Author.

Received 10 May 2019 – Accepted 26 February 2020

http://dx.doi.org/10.4995/agt.2020.11807


F. Barragán, A. Santiago-Santos and J. F. Tenorio

1. Introduction

A continuum is a nonempty compact connected metric space. By a (discrete)
dynamical system we mean a continuum with a continuous self-surjection. This
class of dynamical systems belongs to the area of topological dynamics, which
is a branch of dynamical systems and topology where the qualitative and as-
ymptotic properties of dynamical systems are studied. In the last 30 years,
dynamical systems had been greatly developed, this is because they are very
useful to model problems of other sciences such as Physics, Biology and Eco-
nomics. Currently, we can find several types of dynamical systems: exact, mix-
ing, weakly mixing, transitive, totally transitive, strongly transitive, chaotic,
minimal and sensitive, see [2, 3, 8, 10, 19, 21, 23, 24, 32].

Concerning hyperspaces theory, given a continuum X, the hyperspaces of X
most studied are: the hyperspace 2X which consists of all the nonempty com-
pact subsets of X; given a natural number n, the hyperspace Cn(X) consisting
of the elements of 2X that have at most n components; and the hyperspace
Fn(X) formed by the elements of 2X which have at most n points. Each of
them is topologized with the Hausdorff metric. These hyperspaces are extendly
studied in continuum theory, see [20, 28, 30].

On the other hand, given a continuum X and a positive integer n, in 1979
[29], the study of quotient spaces of hyperspace was initiated with the introduc-
tion of the space C1(X)/F1(X). Later the space Cn(X)/Fn(X) was defined in
2004 [27]. Subsequently, the space Cn(X)/F1(X) was studied [26]. In 2010 [4],
the first named author of this paper defined the space Fn(X)/F1(X) which is
denoted by SFn(X) and is called the n-fold symmetric product suspension of
the continuum X. Some topological properties of SFn(X) are studied in [4, 6].
Finally, in 2013 [14], the space Fn(X)/Fm(X) is defined (1 ≤ m < n) and
is denoted by SFnm(X). In [14] are studied several properties of this quotient
space. Note that when m = 1, SFnm(X) = SFn(X).

A map (continuous surjection) f : X → X, where X is a continuum, induces
a map on the hyperspace 2X, denoted by 2f : 2X → 2X and defined by 2f (A) =
f(A), for each A ∈ 2X. If n is a positive integer, the induced map to the
hyperspace Cn(X) is the restriction of 2f to Cn(X), and is denoted by Cn(f)
and the induced map to the hyperspace Fn(X) is simply the restriction of 2f
to Fn(X) which is denoted by Fn(f). This last map, Fn(f), induces a map on
the space SFn(X) which is denoted by SFn(f) : SFn(X) → SFn(X) [5, 7].
Thus, the dynamical system (X,f) induces the dynamical systems (2X, 2f ),
(Cn(X),Cn(f)), (Fn(X),Fn(f)) and (SFn(X),SFn(f)).

A line of research consists of analyzing the relationships between the dy-
namical system (X,f) (individual dynamic) and the dynamical systems on the
hyperspaces (2X, 2f ), (Cn(X),Cn(f)), (Fn(X),Fn(f)) and (SFn(X),SFn(f))
(collective dynamic). In 1975 [9], the study of this line of research began, and
nowadays there are a lot of results in the literature, for instance in [1, 3, 12,
13, 17, 18, 19, 25, 31, 32, 34]. It is important to note that recently, in 2016 [8],

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 18



Dynamic properties of the dynamical system (SFnm(X),SF
n
m(f))

the relationships between the dynamical systems (X,f), (Fn(X),Fn(f)) and
(SFn(X),SFn(f)) were investigated.

Let n and m be two integers such that n > m ≥ 1 and let X be a con-
tinuum. Note that the function Fn(f) induces another map on the space
SFnm(X) which is denoted by SF

n
m(f) : SF

n
m(X) → SF

n
m(X) [15]. In this

paper, we introduce the dynamical system (SFnm(X),SF
n
m(f)) and we investi-

gate some relationships between the dynamical systems (X,f), (Fn(X),Fn(f)),
(SFn(X),SFn(f)) and (SFnm(X),SF

n
m(f)) when these systems are: exact,

mixing, weakly mixing, transitive, totally transitive, strongly transitive, chaotic,
irreducible, feebly open and turbulent.

This paper is organized as follows: In Section 2, we recall basic defini-
tions and we introduce some notations. In Section 3, we present properties
related with the transitivity of the dynamical systems (X,f), (Fn(X),Fn(f)),
(SFn(X),SFn(f)) and (SFnm(X),SF

n
m(f)), namely: exact, mixing, weakly

mixing, transitive, totally transitive, strongly transitive and chaotic. Finally,
in Section 4, we review others properties of these dynamical systems, namely:
irreducible, feebly open and turbulent.

2. Preliminaries

The symbols N, Q, R and C denote the set of positive integers, rational
numbers, real numbers and complex numbers, respectively. A continuum is a
nonempty compact connected metric space. A continuum is said to be non-
degenerate if it has more than one point. A subcontinuum of a space X is a
continuum contained in X. Given a continuum X, a point a ∈ X and � > 0,
V�(a) denotes the open ball with center a and radius �. A map is a continuous
function. We denote by IdX the identity map on the continuum X.

Given a continuum X and a positive integer n, we consider the hyperspaces
of X, 2X = {A ⊆ X | A is closed and nonempty} and Fn(X) = {A ∈ 2X |
A has at most n points}. We topologize these sets with the Hausdorff metric
[30, (0.1)]. The hyperspace Fn(X) is the n-fold symmetric product of X [11].

Given a finite collection U1,U2, . . . ,Um of nonempty subsets of X, with
〈U1,U2, . . . ,Um〉 we denote the following subset of 2X:{

A ∈ 2X | A ⊆
m⋃
i=1

Ui and A∩Ui 6= ∅, for each i ∈{1, 2, . . . ,m}

}
.

The family:

{〈U1,U2, . . . ,Ul〉 | l ∈ N and U1,U2, . . . ,Ul are open subsets of X}

forms a basis for a topology on 2X called the Vietoris topology [30, (0.11)].
It is well known that the Vietoris topology and the topology induced by the
Hausdorff metric coincide [30, (0.13)]. For those who are interested in learning
more about these topics can see [20, 28, 30].

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 19



F. Barragán, A. Santiago-Santos and J. F. Tenorio

Notation 2.1. Let X be a continuum, let n and m be positive integers, and let
U1,U2, . . . ,Um be a finite family of open subsets of X. By 〈U1,U2, . . . ,Um〉n
we denote the intersection 〈U1,U2, . . . ,Um〉∩Fn(X).

Given two integers n and m such that n > m ≥ 1, SFnm(X) denotes the
quotient space Fn(X)/Fm(X) obtained by shrinking Fm(X) to a point in
Fn(X), with the quotient topology [15]. Here, we denote the quotient map
by qm : Fn(X) →SFnm(X) and qm(Fm(X)) by FmX . Thus:

SFnm(X) = {{A} | A ∈Fn(X) \Fm(X)}∪{F
m
X }.

Note that, if m = 1, then SFn1 (X) = SFn(X) (see [4]).

Remark 2.2. The space SFnm(X)\{FmX } is homeomorphic to Fn(X)\Fm(X),
using the appropriate restriction of qm.

Let n be a positive integer and let X be a continuum. If f : X → X is a
map, we consider the induced map of f on the n-fold symmetric product of X,
Fn(f) : Fn(X) →Fn(X), defined by Fn(f)(A) = f(A), for all A ∈Fn(X) [28,
1.8.23]. Also, given two integers n and m such that n > m ≥ 1, we consider
the function SFnm(f) : SF

n
m(X) →SF

n
m(X) given by:

SFnm(f)(χ) =
{
qm(Fn(f)(q−1m (χ))), if χ 6= FmX ;
FmX , if χ = F

m
X ,

for each χ ∈SFnm(X).
Note that, by [16, 4.3, p. 126], SFnm(f) is continuous. Moreover, diagram 1

is commutative, that is, qm ◦Fn(f) = SFnm(f) ◦qm.

-Fn(X) Fn(X)
Fn(f)

?
SFnm(X)

qm

?
SFnm(X)

qm

-
SFnm(f)

Diagram 1

Note that if m = 1, then SFn1 (f) = SFn(f) (see [5]).
Now, by diagram (∗) from [8, p. 457] and diagram 1, the maps SFn(f) and

SFnm(f) are related under the diagram 2, where q : Fn(X) → SFn(X) is the
quotient map.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 20



Dynamic properties of the dynamical system (SFnm(X),SF
n
m(f))

SFn(X)
SFn(f) // SFn(X)

Fn(X)

qm

��

q

OO

Fn(f) //Fn(X)

qm

��

q

OO

SFnm(X)
SFnm(f) // SFnm(X)

Diagram 2

Observe that SFn(f) ◦q = q ◦Fn(f).

On the other hand, in this paper a dynamical system is a pair (X,f), where
X is a nondegenerate continuum and f : X → X is a map. Given a dynamical
system (X,f), define f0 = IdX and for each k ∈ N, let fk = f ◦fk−1. A point
p ∈ X is a periodic point in (X,f) provided that there exists k ∈ N such that
fk(p) = p. The set of periodic points of (X,f) is denoted by Per(f). Given
x ∈ X, the orbit of x under f is the set O(x,f) = {fk(x) | k ∈ N ∪{0}}.
Finally, a subset K of X is said to be invariant under f if f(K) = K.

Let (X,f) be a dynamical system. We say that (X,f) is:

(1) exact if for each nonempty open subset U of X, there exists k ∈ N such
that fk(U) = X;

(2) mixing if for every pair of nonempty open subsets U and V of X, there
exists N ∈ N such that fk(U) ∩V 6= ∅, for every k ≥ N;

(3) weakly mixing if for all nonempty open subsets U1,U2,V1 and V2 of X,
there exists k ∈ N such that fk(Ui) ∩Vi 6= ∅, for each i ∈{1, 2};

(4) transitive if for every pair of nonempty open subsets U and V of X, there
exists k ∈ N such that fk(U) ∩V 6= ∅;

(5) totally transitive if (X,fs) is transitive, for all s ∈ N;
(6) strongly transitive if for each nonempty open subset U of X, there exists

s ∈ N such that X =
⋃s
k=0 f

k(U);
(7) chaotic if it is transitive and Per(f) is dense in X;
(8) irreducible if the only closed subset A ⊆ X for which f(A) = X is A = X;
(9) feebly open (or semi-open) if for every nonempty open subset U of X, there

is a nonempty open subset V of X such that V ⊆ f(U);
(10) turbulent if there are compact nondegenerate subsets C and K of X such

that C ∩K has at most a point and K ∪C ⊆ f(K) ∩f(C).

Inclusions between some classes of dynamical systems, which are considered
here, are showed in diagram 3. An arrow means inclusion; this is, the class of

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 21



F. Barragán, A. Santiago-Santos and J. F. Tenorio

dynamical system above is contained in the class of dynamical system below.
For some of these inclusions see, for instance, [21, 22].

Exact // Mixing

��
Weakly mixing

��
Totally transitive

��

Strongly transitive

vv

Irreducible

ww

��
Chaotic // Transitive

��

Feebly open

Surjective

Diagram 3

By diagram 3 and [5, Theorem 3.2], we have the following result (compare
with [8, Lemma 2.3]):

Lemma 2.3. Let (X,f) be a dynamical system and n and m be integers such
that n > m ≥ 1. Let N be one of the following classes of dynamical systems:
exact, mixing, weakly mixing, transitive, totally transitive, strongly transitive,
chaotic, and irreducible. If (X,f) ∈ N, then f,Fn(f), SFn(f) and SFnm(f)
are surjective.

Let n be an integer greater than or equal to two and let (X,f) be a dy-
namical system. Observe that F1(X) is a subcontinuum of Fn(X) such that
F1(X) is invariant under Fn(f). In Section 4 of [8] the authors defined and
studied the dynamical system (SFn(X),SFn(f)). Similarly, given an integer
m such that n > m ≥ 1, Fm(X) is also an invariant subcontinuum of Fn(X)
under Fn(f). Thus, by [8, Remark 3.1], we can define the dynamical system
(SFnm(X),SF

n
m(f)).

3. Dynamical properties related to transitivity of
(SFnm(X),SF

n
m(f))

Arguing as in [8, Proposition 4.1] and considering diagram 1, we have the
following result.

Proposition 3.1. Let (X,f) be a dynamical system and let n,m ∈ N be such
that n > m ≥ 1. Then, for each k,s ∈ N, the following holds:
(a) (Fn(f))k(A) = fk(A), for every A ∈Fn(X).
(b) qm ◦ (Fn(f))k = (SFnm(f))k ◦qm.
(c) ((Fn(f))s)k = (Fn(f))sk.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 22



Dynamic properties of the dynamical system (SFnm(X),SF
n
m(f))

(d) qm ◦ ((Fn(f))s)k = ((SFnm(f))s)k ◦qm.

Let (X,d) be a continuum and let f : X → X be a map. Recall that f is an
isometry if d(x,y) = d(f(x),f(y)), for each x,y ∈ X.

Theorem 3.2. Let (X,f) be a dynamical system and let n,m ∈ N be such that
n > m ≥ 1. If f is an isometry, then the dynamical system (SFnm(X),SF

n
m(f))

is not transitive.

Proof. Suppose that f is an isometry and that (SFnm(X),SF
n
m(f)) is tran-

sitive. Let x1, x2, . . . ,xm+1 ∈ X be such that xi 6= xj, for each i,j ∈
{1, . . . ,m + 1} with i 6= j. Let r = min{d(xi,xj) : i,j ∈{1, . . . ,m + 1}, i 6= j},
where d is the metric of X. For each i ∈ {1, . . . ,m + 1}, we put Ui = Vr

4
(xi).

Observe that U1, . . . ,Um+1 are nonempty open subsets of X such that xi ∈ Ui,
for each i ∈{1, . . . ,m + 1} and Ui∩Uj = ∅, for each i,j ∈{1, . . . ,m + 1} with
i 6= j. Moreover, we consider V1, . . . ,Vm+1 nonempty open subsets of X such
that

⋃m+1
i=1 Vi ⊆ U1 and Vi ∩Vj = ∅, for each i,j ∈{1, . . . ,m + 1} with i 6= j.

It follows that 〈U1, . . . ,Um+1〉n is a nonempty open subset of Fn(X) such that
〈U1, . . . ,Um+1〉n∩Fm(X) = ∅ and 〈V1, . . . ,Vm+1〉n∩Fm(X) = ∅. By remark
2.2, we have that qm(〈U1, . . . ,Um+1〉n) and qm(〈V1, . . . ,Vm+1〉n) are nonempty
open subsets of SFnm(X). Since (SF

n
m(X),SF

n
m(f)) is transitive, there exists

k ∈ N such that (SFnm(f))k(qm(〈U1, . . . ,Um+1〉n))∩qm(〈V1, . . . ,Vm+1〉n) 6= ∅.
Hence, by proposition 3.1-(b), we obtain that:

qm((Fn(f))k(〈U1, . . . ,Um+1〉n)) ∩qm(〈V1, . . . ,Vm+1〉n) 6= ∅.

Let B ∈ (Fn(f))k(〈U1, . . . ,Um+1〉n) with qm(B) ∈ qm(〈V1, . . . ,Vm+1〉n).
We consider an element A ∈ 〈V1, . . . ,Vm+1〉n such that qm(A) = qm(B). By
remark 2.2, we have that A = B. Let C ∈ 〈U1, . . . ,Um+1〉n be such that
(Fn(f))k(C) = B. Thus, (Fn(f))k(C) = A. By proposition 3.1-(a), fk(C) =
A. Let c1 ∈ C ∩ U1 and let c2 ∈ C ∩ U2. Hence, d(x1,x2) ≤ d(x1,c1) +
d(c1,c2)+d(c2,x2) <

r
2
+d(c1,c2). This implies that

r
2
< d(c1,c2). On the other

hand, fk(c1),f
k(c2) ∈ fk(C) ⊆

⋃m+1
i=1 Vi ⊆ U1. Thus, d(f

k(c1),f
k(c2)) ≤ r2 .

In consequence, d(fk(c1),f
k(c2)) < d(c1,c2), which is a contradiction to [8,

Remark 4.2]. Therefore, we conclude that (SFnm(X),SF
n
m(f)) is not transitive.

�

The proof of the following result is obtained from theorem 3.2 and diagram
3.

Theorem 3.3. Let (X,f) be a dynamical system and let n,m ∈ N be such
that n > m ≥ 1. Let N be one of the following classes of dynamical systems:
exact, mixing, weakly mixing, transitive, totally transitive, strongly transitive,
and chaotic. If f is an isometry, then (SFnm(X),SF

n
m(f)) 6∈N.

We recall that S1 =
{
e2πiθ ∈ C | θ ∈ [0, 1]

}
.

Example 3.4. Let α ∈ R \ Q and let r : S1 → S1 be the map defined by
r(e2πiθ) = e2πi(θ+α), for each θ ∈ [0, 1]. Note that r is an isometry. Let N be

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 23



F. Barragán, A. Santiago-Santos and J. F. Tenorio

one of the following classes of dynamical systems: exact, mixing, weakly mixing,
transitive, totally transitive, strongly transitive, and chaotic. By theorem 3.3,
we obtain that (SFnm(S1),SF

n
m(r)) 6∈N . On the other hand, we have that the

dynamical system (S1,r) is transitive, totally transitive, and strongly transitive
(see [33, p. 261]).

Theorem 3.5. Let (X,f) be a dynamical system and let n,m ∈ N be such that
n > m ≥ 1. Then the following are equivalent:
(1) (X,f) is exact;
(2) (Fn(X),Fn(f)) is exact;
(3) (SFn(X),SFn(f)) is exact;
(4) (SFnm(X),SF

n
m(f)) is exact.

Proof. By [8, Theorem 4.7], we have that (1), (2) and (3) are equivalent.
Now, if (Fn(X),Fn(f)) is exact, then by, [8, Theorem 3.4], we obtain that
(SFnm(X),SF

n
m(f)) is exact. That is, (2) implies (4). Therefore, for complete

the proof it is enough to prove that (4) implies (1).
Suppose that (SFnm(X),SF

n
m(f)) is exact, we prove that (X,f) is exact.

Let U be a nonempty open subset of X. We see that fk(U) = X, for
some k ∈ N. We take U1, . . . ,Um+1 nonempty open subsets of X such that⋃m+1
i=1 Ui ⊆ U and Ui ∩Uj = ∅, for each i,j ∈{1, . . . ,m + 1} with i 6= j. Note

that 〈U1,U2, . . . ,Um+1〉n is a nonempty open subset of Fn(X), and moreover
〈U1,U2, . . . ,Um+1〉n ∩ Fm(X) = ∅. Hence, by remark 2.2, we obtain that
qm(〈U1,U2, . . . ,Um+1〉n) is a nonempty open subset of SFnm(X). Note that
FmX /∈ qm(〈U1,U2, . . . ,Um+1〉n). Thus, by the assumption, there exists k ∈ N
such that:

(SFnm(f))
k(qm(〈U1,U2, . . . ,Um+1〉n)) = SFnm(X).

In consequence, by part (b) from proposition 3.1, we have that:

qm((Fn(f))k(〈U1,U2, . . . ,Um+1〉n)) = SFnm(X).

Let x ∈ X. We take y1,y2, . . . ,ym ∈ X \ {x} such that yi 6= yj, for each
i,j ∈ {1, 2, . . . ,m} with i 6= j, and we define A = {x,y1, . . . ,ym}. Note that
A ∈ Fn(X) \Fm(X). Thus, qm(A) 6= FmX . Since qm(A) ∈ SF

n
m(X), there

exists B ∈ (Fn(f))k(〈U1,U2, . . . ,Um+1〉n) such that qm(B) = qm(A). Hence,
by remark 2.2, we have that B = A. Let C ∈ 〈U1,U2, . . . ,Um+1〉n be such
that (Fn(f))k(C) = B. By proposition 3.1-(a), we deduce that fk(C) = B.
Since A = B and C ⊆ U, it follows that A ⊆ fk(U). Hence, x ∈ fk(U). Thus,
X ⊆ fk(U). This implies that (X,f) is exact. �

As a consequence from theorem 3.5 and diagram 3, we have the following
result.

Corollary 3.6. Let (X,f) be a dynamical system and let n,m ∈ N be such
that n > m ≥ 1. If (X,f) is exact, then (SFnm(X),SF

n
m(f)) is mixing, weakly

mixing, totally transitive and transitive.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 24



Dynamic properties of the dynamical system (SFnm(X),SF
n
m(f))

Corollary 3.7. Let (X,f) be a dynamical system and let n,m ∈ N be such
that n > m ≥ 1. If f is an isometry, then (X,f) is not exact.

Proof. Suppose that f is an isometry. If the dynamical system (X,f) is exact,
then, by theorem 3.5, the dynamical system (SFnm(X),SF

n
m(f)) is exact. How-

ever, by theorem 3.3, we know that the dynamical system (SFnm(X),SF
n
m(f))

is not exact. Therefore, the dynamical system (X,f) is not exact. �

Theorem 3.8. Let (X,f) be a dynamical system and let n,m ∈ N be such that
n > m ≥ 1. Then the following are equivalent:
(1) (X,f) is mixing;
(2) (Fn(X),Fn(f)) is mixing;
(3) (SFn(X),SFn(f)) is mixing;
(4) (SFnm(X),SF

n
m(f)) is mixing.

Proof. From [8, Theorem 4.9], it follows that (1), (2) and (3) are equivalent.
Now, if the system (Fn(X),Fn(f)) is mixing, then by [8, Theorem 3.4], we have
that the system (SFnm(X),SF

n
m(f)) is mixing. Thus, (2) implies (4). Finally,

we prove that (4) implies (1).
Suppose that (SFnm(X),SF

n
m(f)) is mixing, we prove that (X,f) is mixing.

For this end, let U and V be nonempty open subsets of X. We see that there
exists N ∈ N such that fk(U) ∩ V 6= ∅, for every k ≥ N. We consider
nonempty open subsets U1,U2, . . . ,Um+1 and V1,V2, . . . ,Vm+1 of X such that⋃m+1
i=1 Ui ⊆ U,

⋃m+1
i=1 Vi ⊆ V , Ui∩Uj = ∅ for each i,j ∈{1, 2, . . . ,m + 1} with

i 6= j, and Vi ∩Vj = ∅ for each i,j ∈ {1, 2, . . . ,m + 1} with i 6= j. It follows
that 〈U1,U2, . . . ,Um+1〉n and 〈V1,V2, . . . ,Vm+1〉n are nonempty open subset of
Fn(X) such that 〈U1,U2, . . . ,Um+1〉n ∩Fm(X) = ∅ and 〈V1,V2, . . . ,Vm+1〉n ∩
Fm(X) = ∅. Hence, by remark 2.2, we have that:

qm(〈U1,U2, . . . ,Um+1〉n) and qm(〈V1,V2, . . . ,Vm+1〉n)

are open subsets of SFnm(X). Note that FmX /∈ qm(〈U1,U2, . . . ,Um+1〉n). Addi-
tionally, FmX /∈ qm(〈V1,V2, . . . ,Vm+1〉n). Since (SF

n
m(X),SF

n
m(f)) is mixing,

there exists N ∈ N such that for each k ≥ N:

(SFnm(f))
k(qm(〈U1,U2, . . . ,Um+1〉n)) ∩qm(〈V1,V2, . . . ,Vm+1〉n) 6= ∅.

Fix k ≥ N and let χ ∈ qm(〈U1, . . . ,Um+1〉n) satisfying (SFnm(f))k(χ) ∈
qm(〈V1, . . . ,Vm+1〉n). Let A ∈ 〈U1,U2, . . . ,Um+1〉n such that qm(A) = χ
and let B ∈ 〈V1,V2, . . . ,Vm〉n such that (SFnm(f))k(χ) = qm(B). Hence, we
have that (SFnm(f))k(qm(A)) = qm(B). By part (b) from proposition 3.1,
we obtain that qm((Fn(f))k(A)) = qm(B). From remark 2.2, it follows that
(Fn(f))k(A) = B. Again, by part (a) from proposition 3.1, we deduce that
fk(A) = B. We take a ∈ A ∩ U1. This implies that fk(a) ∈ fk(A) ∩ fk(U).
Moreover, fk(a) ∈ B ∩fk(U). Since B ⊆ V , we have that fk(U) ∩V 6= ∅. In
consequence, (X,f) is mixing. �

Using theorem 3.8 and diagram 3, we deduce the following result.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 25



F. Barragán, A. Santiago-Santos and J. F. Tenorio

Corollary 3.9. Let (X,f) be a dynamical system and let n,m ∈ N be such that
n > m ≥ 1. If (X,f) is mixing, then (SFnm(X),SF

n
m(f)) is weakly mixing,

totally transitive and transitive.

The proof of the following result is similar to the proof of the corollary 3.7.

Corollary 3.10. Let (X,f) be a dynamical system. If f is an isometry, then
(X,f) is not mixing.

Theorem 3.11. Let (X,f) be a dynamical system and let n,m ∈ N be such
that n > m ≥ 1. Consider the following statements:
(1) (X,f) is transitive;
(2) (Fn(X),Fn(f)) is transitive;
(3) (SFn(X),SFn(f)) is transitive;
(4) (SFnm(X),SF

n
m(f)) is transitive.

Then (2), (3) and (4) are equivalent, (2) implies (1), (3) implies (1), (4)
implies (1), (1) does not imply (2), (1) does not imply (3), and (1) does not
imply (4).

Proof. By [8, Theorem 4.10], we have that (2) and (3) are equivalent, (2) implies
(1), (3) implies (1), (1) does not imply (2) and (1) does not imply (3). Now, if
(Fn(X),Fn(f)) is transitive, then by [8, Theorem 3.4], (SFnm(X),SF

n
m(f)) is

transitive. Hence, we have that (2) implies (4).
Finally, suppose that (SFnm(X),SF

n
m(f)) is transitive. We prove that the

system (SFn(X),SFn(f)) is transitive. Let Γ and Λ be nonempty open subsets
of SFn(X). Since q−1(Γ) and q−1(Λ) are nonempty open subsets of Fn(X),
then by [19, Lemma 4.2], there exist nonempty open subsets U1,U2, . . . ,Un and
V1,V2, . . . ,Vn of X such that:

〈U1,U2, . . . ,Un〉n ⊆ q−1(Γ) and 〈V1,V2, . . . ,Vn〉n ⊆ q−1(Λ).
We take, for each i ∈{1, 2, . . . ,n}, a nonempty open subset Wi of X such that
Wi ⊆ Ui and for each i,j ∈ {1, 2, . . . ,n}, Wi ∩ Wj = ∅ with i 6= j. Also,
for each i ∈ {1, 2, . . . ,n}, let Oi be a nonempty open subset of X such that
Oi ⊆ Vi and for each i,j ∈{1, 2, . . . ,n}, Oi∩Oj = ∅ with i 6= j. It follows that
〈U1,U2, . . . ,Un〉n and 〈V1,V2, . . . ,Vn〉n are nonempty open subsets of Fn(X)
such that:

〈W1,W2, . . . ,Wn〉n ⊆〈U1,U2, . . . ,Un〉n ⊆ q−1(Γ)
and

〈O1,O2, . . . ,On〉n ⊆〈V1,V2, . . . ,Vn〉n ⊆ q−1(Λ).
Moreover, 〈W1,W2, . . . ,Wn〉n∩Fm(X) = ∅ and 〈O1,O2, . . . ,On〉n∩Fm(X) =
∅. Hence, by remark 2.2, we have that:

qm(〈W1,W2, . . . ,Wn〉n) and qm(〈O1,O2, . . . ,On〉n)
are nonempty open subsets of SFnm(X). Note that FmX /∈ qm(〈W1, . . . ,Wn〉n)
and FmX /∈ qm(〈O1, . . . ,On〉n). Because (SF

n
m(X),SF

n
m(f)) is transitive, there

exists k ∈ N such that:
(SFnm(f))

k(qm(〈W1,W2, . . . ,Wn〉n)) ∩qm(〈O1,O2, . . . ,On〉n) 6= ∅.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 26



Dynamic properties of the dynamical system (SFnm(X),SF
n
m(f))

As a consequence of proposition 3.1-(d), it follows that:

qm((Fn(f))k(〈W1,W2, . . . ,Wn〉n)) ∩qm(〈O1,O2, . . . ,On〉n) 6= ∅.

Let B ∈ (Fn(f))k(〈W1,W2, . . . ,Wn〉n) with qm(B) ∈ qm(〈O1,O2, . . . ,On〉n).
Hence, we consider A ∈ 〈O1,O2, . . . ,On〉n such that qm(A) = qm(B). By
remark 2.2, we obtain that A = B. Thus, it follows that:

(Fn(f))k(〈W1,W2, . . . ,Wn〉n) ∩〈O1,O2, . . . ,On〉n 6= ∅.

Hence, there is an element C ∈ 〈W1,W2, . . . ,Wn〉n such that (Fn(f))k(C) ∈
〈O1,O2, . . . ,On〉n. Then, q(C) ∈ q(〈W1,W2, . . . ,Wn〉n) and q((Fn(f))k(C)) ∈
q(〈O1,O2, . . . ,On〉n). Moreover, since q ◦Fn(f) = SFn(f) ◦ q, we obtain that
(SFn(f))k(q(C))) ∈ q(〈O1,O2, . . . ,On〉n). Also, observe that:

q(〈W1, . . . ,Wn〉n) ⊆ q(q−1(Γ)) ⊆ Γ and q(〈O1,O2, . . . ,On〉n) ⊆ q(q−1(Λ)) ⊆ Λ.

Hence, (SFn(f))k(Γ) ∩ Λ 6= ∅. In consequence, (SFn(X),SFn(f)) is transi-
tive. Since (2) and (4) are equivalent, we obtain that (4) implies (1).

By example 3.4, we note that (1) does not imply (4). �

As a consequence of diagram 3 and theorem 3.11, we have the next result:

Corollary 3.12. Let (X,f) be a dynamical system and let n,m ∈ N be such
that n > m ≥ 1. If (SFnm(X),SF

n
m(f)) is strongly transitive, then the system

(SFn(X),SFn(f)) is transitive.

Theorem 3.13. Let (X,f) be a dynamical system and let n,m ∈ N be such
that n > m ≥ 1. Then the following are equivalent:
(1) (X,f) is weakly mixing;
(2) (Fn(X),Fn(f)) is weakly mixing;
(3) (Fn(X),Fn(f)) is transitive;
(4) (SFn(X),SFn(f)) is weakly mixing;
(5) (SFn(X),SFn(f)) is transitive;
(6) (SFnm(X),SF

n
m(f)) is weakly mixing;

(7) (SFnm(X),SF
n
m(f)) is transitive.

Proof. By [8, Theorem 4.11], we have that (1), (2), (3), (4) and (5) are equiva-
lent. On the other hand, by theorem 3.11, we have that (5) and (7) are equiva-
lent. It follows from diagram 3 that (6) implies (7). Now, if (Fn(X),Fn(f)) is
weakly mixing, then by [8, Theorem 3.4], (SFnm(X),SF

n
m(f)) is weakly mix-

ing. Hence, we have that (2) implies (6). Thus, (7) implies (6). Therefore, (6)
and (7) are equivalent. �

The proof of the corollary 3.14 is similar to the proof of the corollary 3.7.

Corollary 3.14. Let (X,f) be a dynamical system. If f is an isometry, then
(X,f) is not weakly mixing.

Moreover, by corollary 3.12 and theorem 3.13, we obtain:

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 27



F. Barragán, A. Santiago-Santos and J. F. Tenorio

Corollary 3.15. Let (X,f) be a dynamical system and let n,m ∈ N be such
that n > m ≥ 1. If (SFnm(X),SF

n
m(f)) is strongly transitive, then the system

(SFnm(X),SF
n
m(f)) is weakly mixing.

Theorem 3.16. Let (X,f) be a dynamical system and let n,m ∈ N be such
that n > m ≥ 1. Consider the following statements:
(1) (X,f) is totally transitive;
(2) (Fn(X),Fn(f)) is totally transitive;
(3) (SFn(X),SFn(f)) is totally transitive;
(4) (SFnm(X),SF

n
m(f)) is totally transitive.

Then (2), (3) and (4) are equivalent, (2) implies (1), (3) implies (1), (4)
implies (1), (1) does not imply (2), (1) does not imply (3) and (1) does not
imply (4).

Proof. By [8, Theorem 4.12], we have that (2) and (3) are equivalent, (3) implies
(1), (2) implies (1), (1) does not imply (2) and (1) does not imply (3). Now,
if the system (Fn(X),Fn(f)) is totally transitive, then, by [8, Theorem 3.4],
we have that the system (SFnm(X),SF

n
m(f)) is totally transitive. That is, (2)

implies (4). In consequence (3) implies (4).
Now, we prove that (4) implies (3). Suppose that (SFnm(X),SF

n
m(f)) is

totally transitive, we prove that (SFn(X),SFn(f)) is totally transitive. For
this end, let s ∈ N. We see that (SFn(X), (SFn(f))s) is transitive. Let Γ and
Λ be nonempty open subsets of SFn(X). Since q is continuous, q−1(Γ) and
q−1(Λ) are nonempty open subsets of Fn(X). Applying [19, Lemma 4.2], we
can take nonempty open subsets U1,U2, . . . ,Un and V1,V2, . . . ,Vn of X such
that 〈U1,U2, . . . ,Un〉n ⊆ q−1(Γ) and 〈V1,V2, . . . ,Vn〉n ⊆ q−1(Λ). Hence, for
every i ∈ {1, 2, . . . ,n}, we consider a nonempty open subset Wi of X such
that Wi ⊆ Ui and for each i,j ∈ {1, 2, . . . ,n}, Wi ∩ Wj = ∅, when i 6= j.
Moreover, for every i ∈ {1, 2, . . . ,n}, let Oi be a nonempty open subset of
X such that Oi ⊆ Vi and for each i,j ∈ {1, 2, . . . ,n}, Oi ∩ Oj = ∅, when
i 6= j. Observe that 〈U1,U2, . . . ,Un〉n and 〈V1,V2, . . . ,Vn〉n are nonempty open
subsets of Fn(X) with 〈W1,W2, . . . ,Wn〉n ⊆ 〈U1,U2, . . . ,Un〉n ⊆ q−1(Γ) and
〈O1,O2, . . . ,On〉n ⊆ 〈V1,V2, . . . ,Vn〉n ⊆ q−1(Λ). Moreover, 〈W1, . . . ,Wn〉n ∩
Fm(X) = ∅ and 〈O1,O2, . . . ,On〉n ∩ Fm(X) = ∅. Hence, by remark 2.2,
we have that qm(〈W1,W2, . . . ,Wn〉n) and qm(〈O1,O2, . . . ,On〉n) are nonempty
open subsets of SFnm(X). Note that:

FmX /∈ qm(〈W1, . . . ,Wn〉n) and F
m
X /∈ qm(〈O1,O2, . . . ,On〉n).

Since (SFnm(X),SF
n
m(f)) is totally transitive, (SF

n
m(X), (SF

n
m(f))

s) is tran-
sitive. It follows that there exists k ∈ N such that:

((SFnm(f))
s)k(qm(〈W1,W2, . . . ,Wn〉n)) ∩qm(〈O1,O2, . . . ,On〉n) 6= ∅.

Using proposition 3.1-(d), we obtain that:

qm(((Fn(f))s)k(〈W1,W2, . . . ,Wn〉n)) ∩qm(〈O1,O2, . . . ,On〉n) 6= ∅.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 28



Dynamic properties of the dynamical system (SFnm(X),SF
n
m(f))

By remark 2.2, we have that:

(Fn(f))s)k(〈W1,W2, . . . ,Wn〉n) ∩〈O1,O2, . . . ,On〉n 6= ∅.
In consequence, there exists C ∈ 〈W1,W2, . . . ,Wn〉n such that (Fn(f))s)k(C) ∈
〈O1,O2, . . . ,On〉n. Then q(C) ∈ q(〈W1,W2, . . . ,Wn〉n) and q((Fn(f))s)k(C)) ∈
q(〈O1,O2, . . . ,On〉n). Since q ◦Fn(f) = SFn(f) ◦q, we obtain that:

((SFn(f))s)k(q(C))) ∈ q(〈O1,O2, . . . ,On〉n).
Moreover, we note that q(〈W1,W2, . . . ,Wn〉n) ⊆ Γ and q(〈O1,O2, . . . ,On〉n) ⊆
Λ. Hence, q(C) ∈ Γ and ((SFn(f))s)k(q(C))) ∈ Λ. Thus, it follows that
((SFn(f))s)k(Γ) ∩ Λ 6= ∅. Therefore, (3) and (4) are equivalent. In conse-
quence, (4) implies (2), (4) implies (1), and (1) does not imply (4). �

Theorem 3.17. Let (X,f) be a dynamical system and let n,m ∈ N be such
that n > m ≥ 1. Consider the following statements:
(1) (X,f) is strongly transitive;
(2) (Fn(X),Fn(f)) is strongly transitive;
(3) (SFn(X),SFn(f)) is strongly transitive;
(4) (SFnm(X),SF

n
m(f)) is strongly transitive.

Then (2) implies (1), (2) implies (3), (2) implies (4), (3) implies (1), (4)
implies (1), (1) does not imply (2), (1) does not imply (3), and (1) does not
imply (4).

Proof. By [8, Theorem 4.13], we have that (2) implies (1), (2) implies (3), (3)
implies (1), (1) does not imply (2) and (1) does not imply (3). On the other
hand, if (Fn(X),Fn(f)) is strongly transitive, then, by [8, Theorem 3.4], we
have that (SFnm(X),SF

n
m(f)) is strongly transitive. Hence, (2) implies (4).

Also, by example 3.4, we have that (1) does not implies (4).
We prove that (4) implies (1). Suppose that (SFnm(X),SF

n
m(f)) is strongly

transitive. Let U be a nonempty open subset of X. Let U1, . . . ,Un be nonempty
open subsets of X such that

⋃n
i=1 Ui ⊆ U and Ui ∩ Uj = ∅ for each i,j ∈

{1, . . . ,n} with i 6= j. It follows that 〈U1, . . . ,Un〉n is a nonempty open subset
of Fn(X) such that 〈U1, . . . ,Un〉n ∩Fm(X) = ∅. Using remark 2.2, we obtain
that qm(〈U1, . . . ,Un〉n) is a nonempty open subset of SFnm(X). Note that
FmX /∈ qm(〈U1, . . . ,Un〉n). Considering that (SF

n
m(X),SF

n
m(f)) is strongly

transitive, we have that SFnm(X) =
⋃s
k=0(SF

n
m(f))

k(qm(〈U1, . . . ,Un〉n)), for
some s ∈ N. As a consequence from proposition 3.1-(b), it follows that:

SFnm(X) =
s⋃

k=0

qm((Fn(f))k(〈U1, . . . ,Un〉n)).

Finally, we see that X =
⋃s
k=0 f

k(U). Let x ∈ X. We take y1, . . . ,ym ∈ X\{x}
such that yi 6= yj for each i,j ∈{1, . . . ,m} with i 6= j. Let A = {x,y1, . . . ,ym}.
We have that A ∈ Fn(X) \ Fm(X). In consequence, qm(A) ∈ SFnm(X) \
{FmX }. This implies that there exists j ∈ {0, 1, . . . ,s} such that qm(A) ∈
qm((Fn(f))j(〈U1, . . .Un〉n)). Hence, there exists B ∈ (Fn(f))j(〈U1, . . . ,Un〉n)
such that qm(B) = qm(A). Note that, by remark 2.2, A = B. Observe that

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 29



F. Barragán, A. Santiago-Santos and J. F. Tenorio

there exists C ∈ 〈U1, . . . ,Un〉n such that (Fn(f))j(C) = B. Thus, by proposi-
tion 3.1-(a), fj(C) = B. Moreover, since C ⊆ U, it follows that fj(C) ⊆ fj(U).
Then, A ⊆ fj(U). In consequence, x ∈ fj(U). Thus, X ⊆

⋃s
k=0 f

k(U). Hence,
(X,f) is strongly transitive. �

We have the following questions (compare with [8, Question 4.1]).

Questions 3.18. Let (X,f) be a dynamical system and let n,m ∈ N be such
that n > m ≥ 1.

(i) If (SFnm(X),SF
n
m(f)) is strongly transitive, then is (Fn(X),Fn(f))

strongly transitive?
(ii) If (SFnm(X),SF

n
m(f)) is strongly transitive, then is (SFn(X),SFn(f))

strongly transitive?
(iii) If (SFn(X),SFn(f)) is strongly transitive, then is (SFnm(X),SF

n
m(f))

strongly transitive?

In order to prove the theorem 3.20, we have the next result.

Lemma 3.19. Let (X,f) be a dynamical system and let n,m ∈ N be such
that n > m ≥ 1. Suppose that f is a surjective map. Then the following are
equivalent:

(1) Per(f) is dense in X;
(2) Per(Fn(f)) is dense in Fn(X);
(3) Per(SFn(f)) is dense in SFn(X);
(4) Per(SFnm(f)) is dense in SF

n
m(X).

Proof. By [8, Theorem 4.16], we have that (1), (2) and (3) are equivalent. Now,
by [8, Lemma 3.3], we have that (2) implies (4). Therefore, for complete the
proof it is enough to prove that (4) implies (2).

Suppose that Per(SFnm(f)) is dense in SF
n
m(X), we prove that Per(Fn(f))

is dense in Fn(X). For this end, let U be a nonempty open subset of Fn(X).
By [19, Lemma 4.2], there exist nonempty open subsets U1,U2, . . . ,Un of X
such that 〈U1,U2, . . . ,Un〉n ⊆ U. For each i ∈ {1, 2, . . . ,n}, let Wi be a
nonempty open subset of X such that Wi ⊆ Ui and for each i,j ∈{1, 2, . . . ,n},
Wi ∩ Wj 6= ∅, if i 6= j. It follows that 〈W1,W2, . . . ,Wn〉n is a nonempty
open subset of Fn(X) such that 〈W1,W2, . . . ,Wn〉n ⊆ 〈U1,U2, . . . ,Un〉n ⊆
U and 〈W1,W2, . . . ,Wn〉n ∩ Fm(X) = ∅. Hence, by remark 2.2, we have
that qm(〈W1,W2, . . . ,Wn〉n) is a nonempty open subset of SFnm(X). Ob-
serve that FmX /∈ qm(〈W1,W2, . . . ,Wn〉n). Thus, by hypothesis, we obtain
that qm(〈W1,W2, . . . ,Wn〉n) ∩ Per(SFnm(f)) 6= ∅. In consequence, there exist
A ∈ 〈W1,W2, . . . ,Wn〉n and k ∈ N such that (SFnm(f))k(qm(A)) = qm(A).
This implies, by proposition 3.1-(b) that qm((Fn(f))k(A)) = qm(A). Further-
more, by remark 2.2, we have that (Fn(f))k(A) = A. Therefore, there exist
A ∈ U and k ∈ N such that (Fn(f))k(A) = A. Hence, Per(Fn(f)) is dense in
Fn(X). �

Theorem 3.20. Let (X,f) be a dynamical system and let n,m ∈ N be such
that n > m ≥ 1. Then the next propositions are equivalent:

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 30



Dynamic properties of the dynamical system (SFnm(X),SF
n
m(f))

(1) (X,f) is weakly mixing and chaotic;
(2) (Fn(X),Fn(f)) is chaotic;
(3) (SFn(X),SFn(f)) is chaotic;
(4) (SFnm(X),SF

n
m(f)) is chaotic.

Proof. By [8, Theorem 4.17], we have that (1), (2) and (3) are equivalent.
Now, if (Fn(X),Fn(f)) is chaotic, then, by [8, Theorem 3.4], we have that
(SFnm(X),SF

n
m(f)) is chaotic. Thus, (2) implies (4). As a consequence of

lemma 3.19 and theorem 3.11, we conclude that (4) implies (2). �

Arguing as in corollary 3.7, we obtain the following result.

Corollary 3.21. Let (X,f) be a dynamical system. If f is an isometry, then
(X,f) is not chaotic or (X,f) is not weakly mixing.

4. Other dynamical properties of (SFnm(X),SF
n
m(f))

In this section we study irreducible, feebly open and turbulent dynamical
systems.

Theorem 4.1. Let (X,f) be a dynamical system and let n,m ∈ N be such that
n > m ≥ 1. Consider the following statements:
(1) (X,f) is irreducible;
(2) (Fn(X),Fn(f)) is irreducible;
(3) (SFn(X),SFn(f)) is irreducible;
(4) (SFnm(X),SF

n
m(f)) is irreducible.

Then (2) implies (1), (3) implies (1) and (4) implies (1).

Proof. By [8, Theorem 5.1], we obtain that (2) implies (1) and (3) implies (1).
Therefore, it is enough to prove that (4) implies (1).

Suppose that (SFnm(X),SF
n
m(f)) is irreducible and we prove that (X,f)

is irreducible. We take a nonempty closed subset A of X with f(A) = X.
We see that A = X. Note that 〈A〉n is a nonempty closed subset of Fn(X)
such that Fn(f)(〈A〉n) = Fn(X). Thus, qm(Fn(f)(〈A〉n)) = SFnm(X). Hence,
by proposition 3.1-(b), we have that SFn(f)(qm(〈A〉n)) = SFnm(X). Since
qm(〈A〉n) is a nonempty closed subset of SFnm(X) and (SF

n
m(X),SF

n
m(f))

is irreducible, we have that qm(〈A〉n) = SFnm(X). Now, let x ∈ X and we
consider y1, . . . ,ym ∈ X \ {x} such that yi 6= yj for each i,j ∈ {1, . . . ,m}
with i 6= j. Let B = {x,y1, . . . ,ym}. Clearly, B ∈ Fn(X) \Fm(X). Then,
qm(B) ∈ SFnm(X) \ {FmX }. Considering that qm(B) ∈ SF

n
m(X), there exists

an element C ∈ 〈A〉n with qm(C) = qm(B). Using remark 2.2, we obtain
that C = B. Thus, x ∈ A. This implies that X = A. Therefore, (X,f) is
irreducible. �

Questions 4.2. Let (X,f) be a dynamical system and let n,m ∈ N be such
that n > m ≥ 1.
(i) If (X,f) is irreducible, then is (SFnm(X),SF

n
m(f)) irreducible?

(ii) If (Fn(X),Fn(f)) is irreducible, then is (SFnm(X),SF
n
m(f)) irreducible?

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 31



F. Barragán, A. Santiago-Santos and J. F. Tenorio

(iii) If (SFn(X),SFn(f)) is irreducible, then is (SFnm(X),SF
n
m(f)) irredu-

cible?

Theorem 4.3. Let (X,f) be a dynamical system and let n,m ∈ N be such
that n > m ≥ 1 with f a surjective map. Then the following propositions are
equivalent:

(1) (X,f) is feebly open;
(2) (Fn(X),Fn(f)) is feebly open;
(3) (SFn(X),SFn(f)) is feebly open;
(4) (SFnm(X),SF

n
m(f)) is feebly open.

Proof. By [7, Theorem 10.1], we deduce that (1), (2) and (3) are equivalent.
Now, by theorem [7, Theorem 3.3], it follows that (2) and (4) are equivalent. �

Statement (1) in corollary 4.4 is a consequence of diagram 3 and theorem
4.3. Also, statement (2) in corollary 4.4 is a direct consequence of diagram 3.

Corollary 4.4. Let (X,f) a dynamical system and n and m be integers such
that n > m ≥ 1. Then the following propositions hold:
(1) If (X,f) is irreducible, then (SFnm(X),SF

n
m(f)) is feebly open.

(2) If (SFnm(X),SF
n
m(f)) is irreducible, then (SF

n
m(X),SF

n
m(f)) is feebly

open.

Theorem 4.5. Let (X,f) be a dynamical system and let n,m ∈ N be such that
n > m ≥ 1, where f is a surjective map. Consider the following statements:
(1) (X,f) is turbulent;
(2) (Fn(X),Fn(f)) is turbulent;
(3) (SFn(X),SFn(f)) is turbulent;
(4) (SFnm(X),SF

n
m(f)) is turbulent.

Then (1) implies (2), (3) and (4).

Proof. By [8, Theorem 5.6], we have that (1) implies (2) and (3).
Now, suppose that (X,f) is turbulent. We see that (SFnm(X),SF

n
m(f)) is

turbulent. Let K and C be nondegenerate compact subsets of X such that
K ∩C has at most one point and K ∪C ⊆ f(K) ∩f(C). Observe that 〈K〉n
and 〈C〉n are nondegenerate compact subsets of Fn(X). Let Λ = qm(〈K〉n)
and Γ = qm(〈C〉n). This implies that Λ and Γ are nondegenerate compact
subsets of SFnm(X). Next, we see that Λ ∩ Γ has at most one point. We have
two cases:

Case (1): K ∩ C = ∅. In this case, it follows that 〈K〉n ∩ 〈C〉n = ∅.
Moreover, since Fm(K) ⊆〈K〉n and Fm(C) ⊆〈C〉n, we see that FmX ∈ Λ ∩ Γ.

Case (2): K ∩ C = {a}. In this case, we have that 〈K〉n ∩〈C〉n = {{a}}.
Thus, FmX ∈ Λ ∩ Γ. Now, we suppose that χ ∈ (Λ ∩ Γ) \ {F

m
X }. Then, there

exist A ∈ 〈K〉n\Fm(X) and B ∈ 〈C〉n\Fm(X) such that qm(A) = χ = qm(B).
Using remark 2.2, we obtain that A = B. Hence, A ⊆ K∩C. Thus, K∩C has
at least two elements, which is a contradiction. Therefore, Λ ∩ Γ has at most
one point.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 32



Dynamic properties of the dynamical system (SFnm(X),SF
n
m(f))

We prove that Λ∪Γ ⊆SFnm(f)(Λ)∩SF
n
m(f)(Γ). For this end, we consider

χ ∈ Λ ∪ Γ. It follows that, there exists A ∈ 〈K〉n ∪〈C〉n such that qm(A) =
χ. This implies that A ⊆ f(K) ∩ f(C). Hence, A ∈ 〈f(K) ∩ f(C)〉n. In
consequence, qm(A) ∈ qm(〈f(K)〉n) ∩qm(〈f(C)〉n). Since qm(A) = χ, we have
that χ ∈ qm(Fn(f)(〈K〉n)) ∩ qm(Fn(f)(〈C〉n)). By part (b) from proposition
3.1, we obtain that χ ∈ SFnm(f)(qm(〈K〉n)) ∩SF

n
m(f)(qm(〈C〉n)). Thus, χ ∈

SFnm(f)(Λ)∩SF
n
m(f)(Γ). Then, Λ∪Γ ⊆SF

n
m(f)(Λ)∩SF

n
m(f)(Γ). Therefore,

(SFnm(X),SF
n
m(f)) is turbulent. �

Finally, we have the following questions (compare with [8, Questions 5.7]).

Questions 4.6. Let (X,f) be a dynamical system and let n,m ∈ N be such
that n > m ≥ 1.
(i) If (Fn(X),Fn(f)) is turbulent, then is (SFnm(X),SF

n
m(f)) turbulent?

(ii) If (SFn(X),SFn(f)) is turbulent, then is (SFnm(X),SF
n
m(f)) turbulent?

(iii) If (SFnm(X),SF
n
m(f)) is turbulent, then is (X,f) turbulent?

(iv) If (SFnm(X),SF
n
m(f)) is turbulent, then is (Fn(X),Fn(f)) turbulent?

(v) If (SFnm(X),SF
n
m(f)) is turbulent, then is (SFn(X),SFn(f)) turbulent?

Acknowledgements. The authors thanks the referee for his/her valuable
suggestions which improved the paper.

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