On ω-limit sets of non-autonomous discrete dynamical system

Adv. Fixed Point Theory, 6 (2016), No. 3, 287-294 ISSN: 1927-6303

ON ω-LIMIT SETS OF NON-AUTONOMOUS DISCRETE DYNAMICAL SYSTEM

HONGYING SI∗

School of Mathematics and Information Science, Shangqiu Normal University,

Shangqiu, Henan 476000, PR China

Copyright c2016 Hongying Si. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.In this paper we studyω-limit sets of non-autonomous discrete dynamical systems. Some basic concepts are introduced for non-autonomous discrete systems, includingω-limit set, Lyapunov stable set, asymptotically

stable set. We give some sufficient conditions for non-autonomous discrete dynamical systems to have

asymptoti-cally stable sets.

Keywords:Non-autonomous discrete dynamical systems;ω-limit set; Lyapunov stable set; Asymptotically stable set.

2010 AMS Subject Classification:54H20, 37B55.

1.

Introduction

Throughout this paper, _Ndenotes natural number set and let _Z+ =N∪ {0}. LetX be a

topological space, f_n:X→Xfor eachn∈_Nbe a continuous map and f1,∞denotes the sequence (f₁,f₂,· · ·,f_n,· · ·). The pair (X,f_1,∞)is referred to as a non-autonomous discrete dynamical system [7]. IfX is compact, then(X,f1,∞)is called a compact non-autonomous system. Define

f₁n:= f_n◦f_n−1◦ · · · ◦f₂◦f₁for alln∈_N,

∗_{Corresponding author}

E-mail address: [email protected]

Received March 6, 2016

and f₁0:=id_X, the identity onX. In particular, when f_1,∞is a constant sequence(f,· · ·,f,· · ·), the pair (X,f1,∞) is just classical discrete dynamical system (autonomous discrete dynamical

system)(X,f). The orbit initiated fromx∈X under f_1,∞is defined by the set

γ(x,f1,∞) ={x,f1(x),f

2

1(x),· · ·,f1n(x),· · · }.

Its long-term behaviors are determined by its limit sets.

In past ten years, a large number of papers have been devoted to dynamical properties for non-autonomous discrete systems. Kolyada and Snoha [7] gave definition of topological entropy in non-autonomous discrete systems, Kolyada, Snoha and Trofimchuk [8] discussed minimality of non-autonomous dynamical systems, Kempf [6] and Canovas [2] studiedω-limit sets in

non-autonomous discrete systems respectively. Krabs [9] discussed stability in non-non-autonomous discrete systems, Huang, Wen and Zeng ([4, 5]) studied topological pressure and pre-image entropy of non-autonomous discrete systems, Shi and Chen [13] and Oprocha and Wilczynski [12] discussed chaos in non-autonomous discrete systems respectively.

The concept of asymptotically stable set for classical discrete dynamical system (autonomous discrete dynamical system) was introduced by Block and Coppel [1]. Mimna and Steele [10] discussedω-limit sets and asymptotically stable sets for semi-homeomorphisms, Oprocha [11]

studied asymptotically stable sets in continuous dynamical systems. In this paper we give the notions of ω-limit set and asymptotically stable set for a non-autonomous discrete

sys-tem. Our purpose is to study the properties of asymptotically stable sets for non-autonomous discrete dynamical systems. In particularly, we give necessary and sufficient conditions for non-autonomous discrete systems to have asymptotically stable sets.

2.

Preliminaries

Definition 2.1. Let(X,f1,∞)be a non-autonomous discrete system. For every x∈X and m∈Z+,

the set γm(x,f1,∞) ={f1n(x):n≥m} is called positive orbit through x starting at time m. If

Definition 2.2. Let (X,f1,∞) be a non-autonomous discrete system and let x ∈ X . Define ω(x,f1,∞) as the set of limit points of the orbit γ(x,f1,∞), i.e., ω(x,f1,∞) =

T

m∈Z+

γm(x,f1,∞),

whereγm(x,f1,∞)denotes the closure ofγm(x,f1,∞).

Definition 2.3. [8]Let(X,f1,∞)be a non-autonomous discrete system. Set A⊆X is said to be

invariant if f₁n(A)⊆A for every n∈_N.

For an autonomous system (X,f), by Block and Coppel [1], ifX is a compact space, then

ω(x,f)is invariant for everyx∈X. However, for a non-autonomous system(X,f1,∞), we have ω(x,f1,∞)can not be invariant for somex∈X. We give the following example which is from

[6] to showω(x,f1,∞)is not invariant.

Example 2.1. Let X = [0,1], fn:[0,1]→[0,1]be a sequence of continuous maps and

f_n(x) =

 



1− 1

n+1, for x∈X and n even, 1

n+1, for x∈X and n odd,

for every n∈_N. Thenω(0,f1,∞)is not invariant.

From the definition of f_n(x), we have

f₁n(0) =

 



1

n+1, fornodd,

n

n+1, forneven.

Hence,ω(0,f1,∞) ={0,1}. Since f11(ω(0,f1,∞)) ={0,1₂},ω(0,f1,∞)is not invariant.

Definition 2.4. [13]Let (X,f_1,∞) be a non-autonomous discrete system. f_1,∞ is said to be k-periodic discrete system if there exists k ∈_N such that f_n+k(x) = fn(x) for every x∈X and n∈_N.

Let(X,f_1,∞)be ak-periodic discrete system for ak∈_N. Defineg=: f_k◦f_k−1◦ · · · ◦f₁, we say that(X,g)is induced an autonomous discrete system byk-periodic discrete system(X,f1,∞).

Definition 2.5. [3]Let X be a topological space and {Y_i}i∈I be a family of subsets of X . The family{Y_i}i∈I has the finite intersection property if, for every finite subset J of I, the intersection

T

j∈J

Theorem 2.1. [3]Let X be a metric space and let K be a compact set in X and C be a closed set in X with K∩C= /0. Then there exist two open sets U and V in X , with K⊆U,C⊆V and U∩V = /0.

3.

Asymptotically stable sets of non-autonomous discrete dynamical

sys-tems

In this section, we assume that X is a compact metric space, and we will discuss asymp-totically stable sets of non-autonomous discrete system(X,f_1,∞).

Definition 3.1. Let X be a compact metric space and(X,f_1,∞)be a non-autonomous discrete system. A is a nonempty closed set in X .

(1): A is said to be Lyapunov stable if for each open set U containing A there exists an open set V containing A such thatγ(x,f1,∞)⊆U for every x∈V .

(2): A is said to be asymptotically stable if A is Lyapunov stable and there exists an open set U₀containing A such thatω(x,f1,∞)⊆A for every x∈U0.

Proposition 3.1. Let (X,f1,∞) be a non-autonomous discrete system, where X is a compact

metric space. Let A⊆X be a Lyapunov stable set of(X,f_1,∞). Then A is invariant, i.e., f₁n(A)⊆

A for every n∈_N.

Proof. Suppose that set Ais not invariant. Then there exists n0∈Nsuch that f1n0(A)*A.

Furthermore, there exists a point x₀∈A such that fn0

1 (x0)∈/ A. Since X is a compact metric

space andAis closed, it follows thatX is a Hausdorff space andAis a compact subset inX. By Theorem 2.1, there exist an open neighborhoodU₁of fn0

1 (x0)and an open neighborhoodU2of

Asuch thatU1∩U2= /0, which implies f₁n0(x0)∈/U2. Hence, for every open setV containingA,

we havex₀∈A⊆V and fn0

1 (x0)∈/U2, which impliesγ(x0,f1,∞)*U2. This is a contradiction.

Theorem 3.1. Let(X,f1,∞)be a non-autonomous discrete system, where X is a compact metric

space, and let subset A in X is an asymptotically stable set in (X,f_1,∞). Then there exists

set P={n₁(U),n₂(U),· · ·,n_p(U)}, where n₁(U),n₂(U),· · ·, n_p(U)∈_N, such that for every x∈U₀, there exists a positive integer m∈P satisfying f₁n(f₁m(x))∈U for every n∈_N.

Proof.LetAbe an asymptotically stable set. Then there exists an open neighborhoodW ofA such thatω(x,f1,∞)⊆Afor everyx∈W. SinceAandX\W are two closed subsets ofX andX

is a compact metric space, thenAandX\W are two compact subsets. Furthermore, there exists an open setU₀ofX satisfyingA⊆U₀⊆U₀⊆W.

Notice thatω(x,f1,∞)⊆Afor every pointx∈U0. Take any open neighborhoodU ofA, we

may also assume thatU⊆U₀. There exists an open neighborhoodV ofAsuch thatγ(x,f1,∞) =

{f₁n(x):n=0,1,· · · } ⊆U for every x∈V. Moreover, for every x∈U0, there exists a n=

n(x)∈_Nsuch that f₁n(x)∈V becauseω(x,f1,∞)⊆A. As f1n(x)is continuous, there exists an

open neighborhoodW_x ofx such that f₁n(W_x)⊆V. SetU₀ is compact because X is compact, and family{W_x}is its open cover. Hence, we may choose finite subcover{W_x1,W_x2,· · ·,W_xp}

of U₀. Furthermore, for each W_xi, there exists a n_i= n(x_i)∈_N such that fni

1 (Wxi)⊆V for

i=1,2,· · ·,p. TakeP={n(x₁),n(x₂),· · ·,n(x_p)}. Hence, for everyx∈U₀, there existsm∈P such that f₁m(x)∈V. Therefore, we have f₁n(f₁m(x))∈U for everyn∈_N.

Theorem 3.2. Let (X,f1,∞) be a non-autonomous discrete system, where(X,d)is a compact

metric space. Let A be a closed invariant set and U₀be an open neighborhood of A, if for every

open neighborhood U of A, there exists a N=N(U)∈_Nsuch that f₁n(U0)⊆U for all n≥N.

Then A is an asymptotically stable set.

Proof. Firstly, we show that Ais Lyapunov stable. Suppose that Ais not Lyapunov stable. Then there exists an open neighborhoodU ofA satisfying for every open setV containingA, there exists a pointx∈V such thatγ(x,f1,∞)*U.

SinceU₀ be an open neighborhood of A, we can take points x_k ∈U₀ such that x_k→x∈A whenk→∞and integers n_k∈_N such that f₁nk(x_k)∈/U for all k∈_N. As f₁n(U0)⊆U for all

Secondly, we proveω(x,f1,∞)⊆Afor everyx∈U0. LetU =Bε(A)whereBε(A) ={x∈X: d(x,A)<ε}. SinceU=Bε(A)is an open neighborhood ofA, then there exists aN=N(U)∈N such that f₁n(U₀)⊆B_ε(A) for every n≥N. Furthermore, we have f₁n(x) ∈B_ε(A) for every x∈U0andn≥N. Moreover, for everyy∈ω(x,f1,∞), there exists an increasing sequence{ni}

such thaty=lim i→∞f

ni

1 (x). Take

(1): ε1= 1. Then there exists a N1 =N(B1(A)) ∈N such that f1ni1(x) ∈B1(A) when

n_i1 ≥N₁;

(2): ε2=1₂. Then there exists aN2=N(B1 2(

A))∈_Nsuch that fni2

1 (x)∈B1₂(A)andni2>ni1

whenn_i2 ≥N₂;

(3): ε3=1₃. Then there exists aN3=N(B1 3(

A))∈_Nsuch that fni3

1 (x)∈B1₃(A)andni3>ni2

whenn_i3 ≥N₃, and so on.

Since{n_ij: j=1,2,· · · }is a subsequence of{n_i:i=1,2,· · · }, it follows that lim j→∞f

n_{i j}

1 (x) =y.

Moreover, f₁ni j(x)∈N(B1

j

(A)), i.e.,d(f₁ni j(x),A)< 1_j. Furthermore,

d(y,A)≤d(y,f₁ni j(x)) +d(f₁ni j(x),A).

SinceAis a closed set ofX, thus, when j→∞, we havey∈A. Hence,ω(x,f1,∞)⊆Afor every

x∈U0.

Theorem 3.3. Let(X,f1,∞)be a non-autonomous discrete system, where X is a compact metric

space . Let A be a closed invariant set and there exists an open set V containing A such that

(1): f₁n(V)⊆ f₁n−1(V)⊆V for every n∈_N; (2): T

n∈Z+

f₁n(V)⊆A.

Then A is an asymptotically stable set.

Proof. Since X is a compact space, thus V is a compact subset of X. Moreover, f₁n is a continuous map for every n∈_N. Hence, f₁n(V) is a compact subset of X for every n∈_Z+.

By condition(1), the compact sets f₁n(V)form a decreasing sequence. By Definition 2.5, the family{f₁n(V)}n∈Z+has the finite intersection property. Furthermore, we have

T

n∈Z+

such that f₁n(U₀)⊆U for alln≥N. Therefore, by Theorem 3.2,Ais an asymptotically stable set of(X,f1,∞).

Theorem 3.4. Let(X,f_1,∞)be a k-periodic discrete system, g= f_k◦f_k−1◦ · · · ◦f₁, (X,g)is its induce autonomous discrete system. If A is an asymptotically stable set of(X,f1,∞), then A is

an asymptotically stable set of(X,g).

Proof.Firstly, we showAis Lyapunov stable in(X,g). LetU be any open set which contain-ingA. SinceAis an asymptotically stable set of(X,f1,∞), it follows that there exists an open set

V containingAsuch thatγ(x,f1,∞)⊆U for everyx∈V. As (X,f1,∞)be ak-periodic discrete

system and g= f_k◦ f_k−1◦ · · · ◦ f1= f₁k, we have fn+k(x) = fn(x) for every x∈X. Further-more,gm(x) = (f₁k)m(x) = f₁mk(x). Moreover, for everyx∈V, γ(x,g) ={x,g(x),g2(x),· · · }= {x,f₁k(x),f₁2k(x),· · · }, thus,γ(x,g)⊆γ(x,f1,∞). Hence, we haveγ(x,g)⊆U for everyx∈V.

This showsAis Lyapunov stable in(X,g).

Secondly, we prove that there exists an open setU0 containing Asuch that ω(x,g)⊆Afor

everyx∈U₀. SinceAis an asymptotically stable set of(X,f_1,∞), then there exists an open set U0containingAsuch thatω(x,f1,∞)⊆Afor everyx∈U0. Moreover, for everym∈N, we have γm(x,g)⊆γm(x,f1,∞). Furthermore, we have

T

m∈Z+

γm(x,g)⊆ T m∈Z+

γm(x,f1,∞), which implies ω(x,g)⊆ω(x,f1,∞). Hence,ω(x,g)⊆Afor every x∈U0. This shows Ais an asymptotically

stable set of(X,g). Conflict of Interests

The authors declare that there is no conflict of interests.

Acknowledgements. The work was supported by the Natural Science Foundation of Henan Province(162300410075), PR China. and by the National Natural Science Foundation of China (11401363).

REFERENCES

[1] L.S. Block and W.A. Coppel, Dynamics in One Dimension.Lecture Notes in Mathematics, 1513, Springer

Verlag, Berlin, 1992.

[3] R. Engelking,General Topology.PWN, Warszawa, 1977.

[4] X. Huang, X. Wen and F. Zeng, Topological pressure of nonautonomous dynamical systems, Nonlinear

Dynam. Sys Theory.8(2008), 43-48.

[5] X. Huang, X. Wen and F. Zeng, Pre-image entropy of nonautonomous dynamical systems, Jrl. Syst. Sci.

Complexity.21(2008), 441-445.

[6] R. Kempf, OnΩ-limit sets of discrete-time dynamical systems,J. Difference Eq. Appl.8(2002), 1121-1131. [7] S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput.

Dynam.4(1996), 205-233.

[8] S. Kolyada, L. Snoha and S. Trofimchuk, On minimality of nonautonomous dynamical systems,Nonlinear

Oscil.7(2004), 83-89.

[9] W. Krabs, Stability and controllability in non-autonomous time-discrete dynamical systems,J. Difference Eq.

Appl.8(2002), 1107-1118.

[10] R.A. Mimna and T.H. Steele, Asymptotically stable sets for semi-homeomorphisms, Nonlinear Anal 59 (2004), 849-855.

[11] P. Oprocha, Topological approach to chain recurrence in continuous dynamical systems,Opuscula Math.25 (2005), 261-268.

[12] P. Oprocha and P. Wilczynski, Chaos in nonautonomous dynamical systems,An. St. Ovidius Constanta.17 (2009), 209-221.