@ Appl. Gen. Topol. 15, no. 2(2014), 175-181doi:10.4995/agt.2014.1841
c© AGT, UPV, 2014

Lifting dynamical properties to hyperspaces and

hyperspace suspension

Dania Masood and Pooja Singh

Department of Mathematics, University of Allahabad, Allahabad - 211 002, India. (dpub@pphmj.com,

poojasingh 07@rediffmail.com)

Abstract

For a dynamical system (X, f), the passage of various dynamical pro-
perties such as transitivity, total transitivity, weakly mixing, mixing,
topological exactness, topological conjugacy, to the hyperspace C(X)
of X consisting of nonempty closed connected subsets of X, and also
to the hyperspace suspension HS(X) of X, have been considered and
studied.

2010 MSC: 54H20; 37B99.

Keywords: transitivity; total transitivity; weakly mixing; mixing; topolog-
ical exactness; topological conjugacy; Vietoris topology, hyper-
space suspension.

1. Introduction

Throughout the paper, (X,f) denotes a dynamical system, where X is a
compact metric space and f is a selfmap on X. By a map, we mean a continuous
map. The symbol N denotes the set of positive integers.

Let (X,f) be a dynamical system. Then f is called (i) transitive if for
a pair of nonempty open sets U, V of X, there exists an n ∈ N, such that
fn(U)∩V 6= φ, (ii) totally transitive if the map fk is transitive, for each k ∈ N,
(iii) weakly mixing if for pairs of nonempty open sets U1, U2 and V1, V2 of X,
there exists an n ∈ N, such that fn(Ui) ∩ Vi 6= φ, for i = 1, 2, (iv) mixing
if for a pair of nonempty open sets U, V of X, there exists an N ∈ N, such
that fn(U) ∩ V 6= φ, n ≥ N, (v) topologically exact if for an open set U of X,
there exists an n ∈ N, such that fn(U) = X. It is known that, if f is weakly

Received 7 November 2013 – Accepted 15 March 2014

http://dx.doi.org/10.4995/agt.2014.1841


D. Masood and P. Singh

mixing and U1,U2, . . . ,Un, V1,V2, . . . ,Vn are nonempty open sets of X, then
there exists a k ≥ 1 such that fk(Ui) ∩ Vi 6= φ, for i = 1,2, . . . ,n [12]. The
dynamical system (X,f) is said to possess a dynamical property P , if f has P .

Two dynamical systems (X,f) and (Y,g) are said to be topologically conju-
gate if there exists a homeomorphism ϕ from X to Y such that g ◦ ϕ = ϕ ◦ f.

By 2X, we denote the set consisting of all nonempty closed subsets of the
space X. On it,

(i) the upper Vietoris topology τU , has BU ≡ {< U1, . . . ,Um >: U1, . . . ,Um
are open sets of X}, where < U1, . . . ,Um > = {A ∈ 2

X : A ⊆
⋃m

i=1 Ui}, as a basis,
(ii) the lower Vietoris topology τL, has BL ≡ {< U

′

1, . . . ,U
′

m >: U
′

1, . . . ,U
′

m

are open sets of X}, where < U′1, . . . ,U
′

m > = {A ∈ 2
X : A ∩ U′i 6=

φ, i = 1, . . . ,m}, as a basis, and
(iii) the Vietoris topology τ, has B ≡ {< V1, . . . ,Vn >: V1, . . . ,Vn are open

sets of X}, where < V1, . . . ,Vn >= {A ∈ 2
X : A ⊆

⋃n
i=1 Vi, A ∩ Vi 6=

φ, i = 1, . . . ,n}, as a basis.

The hyperspace C(X), denotes the subspace of 2X consisting of nonempty
closed connected subsets of X. We shall write C(X) to mean (C(X),τ).
The upper and lower Vietoris topologies on C(X) are specified by writing
(C(X),τU ) and (C(X),τL), respectively. For nonempty open sets U1, . . . ,Un
of X and U = Ui, for some i ≤ n, when m ≥ 1, we may write, as in [1],
< U1, . . . ,Un > alternatively in the form < U1, . . . ,Un,U, . . . ,U

︸ ︷︷ ︸

m

>. For a

nonempty basic open set U = < U1, . . . ,Um > of C(X), by
⋃
U we mean

⋃m
i=1 Ui.
Observe that X is embedded into C(X) by the embedding x 7→ {x}. Also,

every continuous selfmap f on X, induces in a natural way, a selfmap f′ on 2X

[2]. The hyperspace C(X ) is invariant with respect to f′, and hence we obtain
a dynamical system (C(X), f̄), where f̄ = f′|C(X).

For k ∈ N, the space Fk(X), denotes the collection of those nonempty closed
subsets of X that consist of atmost k elements. The set F1(X) consisting
of all singletons in X, is a closed subspace of C(X), and the quotient space
C(X)/F1(X), denoted by HS(X ), is known as the hyperspace suspension of
X [10]. Further, for a dynamical system (X,f), the pair (HS(X ), HS(f ))
constitutes a dynamical system with the evolution map HS(f ) sending [A] ∈
HS(X) to (qX ◦ f ◦ q

−1
X

)([A]), where qX denotes the cannonical projection from
C(X ) to HS(X ) [5]. That HS(f ) is continuous follows from [4, 4.3, p. 146]. It
is pertinent to mention that HS taking X to HS(X ) and f to HS(f ) describes
a covariant functor from the category of topological spaces to itself.

A brief exposition of our work is as follows. If (X,f) is transitive, then
(
(C(X),τU ), f̄

)
is transitive. However,

(
(C(X),τL), f̄

)
is transitive if X is

pathconnected and f is weakly mixing. The same is the state of affairs so far
as the total transitivity and weakly mixing are concerned. It is known that
the transitivity and total transitivity do not necessarily lift to C(X) endowed

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 176



Lifting dynamical properties to hyperspaces and hyperspace suspension

with the Vietoris topology. For a dynamical system (X,f) with f mixing,
it is obtained that f̄ on (C(X),τL) continues to be mixing provided X is
pathconnected. All these constitute the contents of Section 3. Section 4 is
devoted to the passage of these dynamical properties to the dynamical system
(HS(X),HS(f)) from the dynamical system (X,f). It has been obtained that
all these properties get lifted smoothly. Finally, it is obtained that topological
conjugacy is preserved under the hyperspace suspension.

We begin with the preliminaries in Section 2, wherein, certain results scat-
tered across various references are stated that we subsequently use.

2. Preliminaries

A topological graph is a connected compact Hausdorff space G for which there
exists a finite collection of subspaces Ii, i = 1,2, . . . ,s such that G =

⋃s
i=1 Ii,

where each Ii is homeomorphic to a compact interval, and each intersection
Ii ∩Ij, for i 6= j is finite. Equivalently, a graph is a one-dimensional connected,
compact polyhedron. Below we state a result related to the lift of transitivity
to the hyperspace C(G) of a dynamical system (G,f), where G is a graph and
f is transitive.

Proposition 2.1 ([9, Cor. 29]). For a dynamical system (G,f), the map f̄ is
never transitive.

Remark 2.2. Note that the map f̄ has been denoted by f̃ in [9].

Denoting by f̂ the map induced on K(X) consisting of all nonempty compact
subsets of X, we have the following:

Proposition 2.3 ([7, Lemma 5]). For a dynamical system (X,f), the following
are equivalent:

(1) f is topologically exact.

(2) f̂ is topologically exact.

Remark 2.4. In [7], the map f̂ has been denoted by f̄, which we have used for
the map induced on the hyperspace C(X ).

Proposition 2.5 ([9, Lemma 4]). If dynamical systems (X,f) and (Y,g) are
topologically conjugate, then the same holds for the induced dynamical systems

(2X,f′) and (2Y ,g′), and also for (C(X), f̄) and (C(Y), ḡ).

Remark 2.6. In [9], the maps f′,g′, f̄ and ḡ have been denoted by f̄, ḡ, f̃ and
g̃, respectively.

Proposition 2.7 ([8]). Topological transitivity is preserved under topological
conjugacy.

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D. Masood and P. Singh

3. Dynamical properties and C(X)

Theorem 3.1. Let (X,f) be a dynamical system with f transitive. Then the
induced map f̄ on (C(X),τU ) is transitive.

Proof. Consider a pair of nonempty basic open sets U and V of C(X). Then
⋃
U and

⋃
V constitute a pair of nonempty open sets in X. Since f is transitive,

there is a natural number l such that for some y ∈
⋃
U, fl(y) ∈

⋃
V. That

f̄l(U) intersects V follows by noting that {y} ∈ U and f̄l({y}) ∈ V. �

Proposition 2.1 shows that there is no dearth of dynamical systems whose
evolution maps are transitive possessing induced evolution maps which fail to
be transitive. Below we provide an example for the sake of illustration.

Example 3.2. The irrational rotation Rα : S
1 → S1 defined by

Rα(θ) = θ + α, θ ∈ [0,2π),

where α is an irrational number, and the multiplicative group S1 is identified
with the additive group [0,2π) mod 2π, is a transitive map [3, 11]. Let Rα
be the map induced on C(S1), which is homeomorphic to D2 ≡ {(r,θ) | r ∈
[0,1] and θ ∈ [0,2π)} [2, 6], via the homeomorphism ψ, sending A ∈ C(S1) to
(1 − lA/2π,mid A) ∈ D

2, where lA and mid A, denote the arclength and the
middle point of A, respectively. The map g : D2 → D2 defined by

g(r,θ) = (r,θ + α), r ∈ [0,1], and θ ∈ [0,2π),

is such that g ◦ ψ = ψ ◦ Rα, making the systems (C(S
1),Rα) and (D

2,g)
topologically conjugate. Since the orbit of no element of D2 under g is dense
in it, the map g is not transitive [8], and hence, by Proposition 2.7, Rα is also
not transitive.

Theorem 3.3. Let (X,f) be a dynamical system with f totally transitive.
Then the induced map f̄ on (C(X),τU ) is totally transitive.

Proof. Since for a natural number n, fn = f̄n, the result follows from Theorem
3.1. �

Theorem 3.4. Let (X,f) be a dynamical system with f weakly mixing. Then
the induced map f̄ on (C(X),τU ) is weakly mixing.

Proof. Consider pairs of nonempty basic open sets U1,V1 and U2,V2 of C(X).
Then

⋃
U1,

⋃
V1 and

⋃
U2,

⋃
V2 are pairs of nonempty open sets of X. Since f

is weakly mixing, there are n ∈ N, y ∈
⋃
U1 and y

′ ∈
⋃
U2 such that f

n(y) ∈
⋃
V1, and f

n(y′) ∈
⋃
V2. That f̄

n(U1) ∩ V1 6= φ and f̄
n(U2) ∩ V2 6= φ, follows

by observing that {y} ∈ U1, f̄
n({y}) ∈ V1, {y

′} ∈ U2 and f̄
n({y′}) ∈ V2. �

Theorem 3.5. Let (X,f) be a dynamical system with f weakly mixing. If X
is pathconnected, then the induced map f̄ on (C(X),τL) is weakly mixing.

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 178



Lifting dynamical properties to hyperspaces and hyperspace suspension

Proof. Consider pairs < U1, . . . ,Um >,< V1, . . . ,Vm >, and < U
′

1, . . . ,U
′

n >,<
V ′1, . . . ,V

′

n > of nonempty basic open sets of C(X). Then the pairs (Ui,Vi), i =
1, . . . ,m, and (U′j,V

′

j ), j = 1, . . . ,n, consist of nonempty open sets of X. Since

f is weakly mixing, there is a k ∈ N, such that fk(Ui) ∩ Vi 6= φ, i = 1, . . . ,m,
and fk(U′j) ∩ V

′

j 6= φ, j = 1, . . . ,n. Thus there are elements xi ∈ Ui, x
′

j ∈ U
′

j

such that fk(xi) ∈ Vi, and f
k(x′j) ∈ V

′

j , i = 1, . . . ,m, j = 1, . . . ,n. Let A

and B be paths containing all xi’s, and all x
′

j’s, respectively. Then A ∈ <

U1, . . . ,Um >, B ∈ < U
′

1, . . . ,U
′

n >, and f̄
k(A) ∈ < V1, . . . ,Vm >, f̄

k(B) ∈ <
V ′1, . . . ,V

′

n >. Hence, the result. �

Because, for a given dynamical system (X,f), the weakly mixing of f implies
that f is transitive and also totally transitive, we have the following:

Corollary 3.6. Let (X,f) be a dynamical system with f weakly mixing. If X
is pathconnected, then the induced map f̄ on (C(X),τL) is transitive and also
totally transitive.

Theorem 3.7. Let (X,f) be a dynamical system with f mixing. Then the
induced map f̄ on (C(X),τU ) is mixing.

Proof. For a pair U,V of nonempty basic open sets of C(X),
⋃
U,

⋃
V constitute

a pair of nonempty open sets of X. Since f is mixing, there is an N ∈ N, such
that for n ≥ N, fn(

⋃
U) intersects

⋃
V, which implies that f̄n(U) intersects

V, for n ≥ N. Hence, the result. �

Theorem 3.8. Let (X,f) be a dynamical system with f mixing. If X is
pathconnected, then the induced map f̄ on (C(X),τL) is mixing.

Proof. Consider a pair < U1, . . . ,Um >,< V1, . . . ,Vm > of nonempty basic
open sets of C(X). Since f is mixing, for each pair (Ui,Vi), i = 1, . . . ,m, of
open sets of X, there exist Ni ∈ N such that f

k(Ui)∩Vi 6= φ, whenever k ≥ Ni.
Set N = max {Ni : i = 1, . . . ,m}. Then f

k(Ui) ∩ Vi 6= φ, for i = 1, . . . ,m,
and k ≥ N. Thus for i = 1, . . . ,m, k ≥ N, there are xik ∈ Ui such that
fk(xik ) ∈ Vi. Let Ak be a path containing all xik ’s. Since f̄

k(Ak) lies in
f̄k(< U1, . . . ,Um >) as well as in < V1, . . . ,Vm >, the result follows. �

4. Dynamical properties and HS(X)

We begin with the following:

Lemma 4.1. For n ∈ N, (HS(f))n ≡ (qX ◦ f
n ◦ q−1

X
).

Proof. The result follows by induction on n. �

Theorem 4.2. Let (X,f) be a dynamical system such that (C(X), f̄) is tran-
sitive. Then the dynamical system (HS(X),HS(f)) is also transitive.

Proof. For a pair U,V of nonempty open sets of HS(X), q−1
X

(U) and q−1
X

(V )
constitute a pair of nonempty open sets of C(X). Since f̄ is transitive, there is
an n ∈ N such that fn(q−1

X
(U)) ∩ q−1

X
(V ) 6= φ. Applying qX and using Lemma

4.1, the result follows. �

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 179



D. Masood and P. Singh

Theorem 4.3. Let (X,f) be a dynamical system such that (C(X), f̄) is to-
tally transitive. Then the dynamical system (HS(X),HS(f)) is also totally
transitive.

Proof. For n ∈ N, Theorem 4.2 and Lemma 4.1 provide the transitivity of
(HS(f))n. Thus, HS(f) is totally transitive. �

Theorem 4.4. Let (X,f) be a dynamical system such that (C(X), f̄) is weakly
mixing. Then the dynamical system (HS(X),HS(f)) is also weakly mixing.

Proof. Consider pairs U1,U2 and V1,V2 of nonempty open sets of HS(X).
Then q−1

X
(U1),q

−1
X

(U2) and q
−1
X

(V1),q
−1
X

(V2) are pairs of nonempty open sets
of C(X). The weakly mixing property of the map f̄ implies the existence of
an n ∈ N, such that fn(q−1

X
(Ui)) ∩q

−1
X

(Vi) 6= φ, for i = 1,2. Now arguing as in
Theorem 4.2, we have the result. �

Theorem 4.5. Let (X,f) be a dynamical system such that (C(X), f̄) is mixing.
Then the dynamical system (HS(X),HS(f)) is also mixing.

Proof. For a pair U,V of nonempty open sets of HS(X), q−1
X

(U) and q−1
X

(V )
constitute a pair of nonempty open sets of C(X). The mixing property of f̄
on C(X) provides an N ∈ N, such that fn(q−1

X
(U)) ∩ q−1

X
(V ) 6= φ, for n ≥ N.

Now proceeding as in the proof of Theorem 4.2, we have the result. �

Theorem 4.6. Let (X,f) be a dynamical system such that (C(X), f̄) is topo-
logically exact. Then the dynamical system (HS(X),HS(f)) is also topologi-
cally exact.

Proof. Let U be a nonempty open set of HS(X). Then q−1
X

(U) is a nonempty
open set of C(X). The topological exactness of f̄ on C(X) yields an n ∈ N,
such that fn(q−1

X
(U)) = C(X). Applying qX and using Lemma 4.1, the result

follows. �

Theorem 4.7. If the dynamical systems (X,f) and (Y,g) are topologically
conjugate, then the dynamical systems (HS(X),HS(f)) and (HS(Y ),HS(g))
are also topologically conjugate.

Proof. Let ϕ be a conjugacy between the pairs (X,f) and (Y,g). Then by
Proposition 2.5, the dynamical systems (C(X), f̄) and (C(Y ), ḡ) are topolog-
ically conjugate, the induced map ϕ̄ being the topological conjugacy. Since ϕ̄
maps F1(X) homeomorphically onto F1(Y ), HS(ϕ̄) : HS(X) → HS(Y ) is the
required topological conjugacy between the dynamical systems (HS(X),HS(f))
and (HS(Y ),HS(g)). �

Acknowledgements. The authors are grateful to their supervisor Professor

K. K. Azad for his valuable guidance and encouragement. The first author

thanks the Department of Science and Technology, New Delhi, for providing

financial support.

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 180



Lifting dynamical properties to hyperspaces and hyperspace suspension

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