# Li-Yorke Sensitivity of Set-Valued Discrete Systems

Authors: Heng Liu; Fengchun Lei; Lidong Wang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Consider the surjective, continuous mapf:X→X and the continuous map f¯ of 𝒦(X)  induced by f, where X is a compact metric space and 𝒦(X) is the space of all nonempty compact subsets of X endowed with the Hausdorff metric. In this paper, we give a short proof that if f¯ is Li-Yoke sensitive, then f is Li-Yorke sensitive. Furthermore, we give an example showing that Li-Yorke sensitivity of f does not imply Li-Yorke sensitivity of f¯.


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## Body

## 1. Introduction

Throughout this paper a dynamical system(X,f) is a pair where X is a compact metric space with metric d and f:X→X is a surjective, continuous map.The idea of sensitivity from the work [1, 2] by Ruelle and Takens was applied to topological dynamics by Auslander and Yorke in [3] and popularized later by Devaney in [4]. A system (X,f) is called ɛ-sensitive if there exists a positive ɛ such that any x∈X is a limit of points y∈X satisfying the condition d(fn(x),fn(y))>ɛ for some positive integer n. According to Li and Yorke (see [5]), a subset S⊂X is a scrambled set (for f), if any different points x and y from S are proximal and not asymptotic; that is,
(1)liminfn→∞d(fn(x),fn(y))=0,limsupn→∞d(fn(x),fn(y))>0.Li-Yoke sensitivity is introduced by Akin and Kolyada in [6]. A system is Li-Yorke sensitive if there exists ɛ>0 such that every x∈X is a limit of points y∈X such that the pair (x,y) is proximal but supn>N{d(fn(x),fn(y))}>ɛ for any N>0, and the positive ɛ is said to be a Li-Yorke sensitive constant of the system. A pair (x,y) is ɛ-Li-Yorke sensitive if the pair  (x,y) is proximal but whose orbits are frequently at least ɛ apart.A dynamical system(X,f) is called spatiotemporal chaotic (see [6] or [7]) if every point is a limit point for points which are proximal to but not asymptotic to it. That is, for any x∈X and any open subset U with x∈U, there is y∈U such that x and y are proximal and not asymptotic. It is easy to see that Li-Yorke sensitivity implies spatiotemporal chaos and sensitivity.Román-Flores [8] and Fedeli [9] studied the interplay of chaos for discrete dynamical systems (individual chaos) with the corresponding set-valued versions (collective chaos). Recall that the map f¯:𝒦(X)→𝒦(X) induced by f on 𝒦(X)={K⊂X:Kisanonemptycompactsubset} is defined by f¯(K)=f(K)={f(x):x∈K}, K∈𝒦(X). Then the pair (𝒦(X),f¯) is a dynamical system with the space 𝒦(X) endowed with the Hausdorff distance:
(2)Hd(K1,K2)=max{sup{d(x1,K2):x1∈K1},sup{d(x2,K2):x2∈K2}},
and K1,K2∈𝒦(X). And various concepts of chaos in set-valued discrete systems have been researched recently (see [10–16]).In this paper, we discuss the relationship between Li-Yorke sensitivity off and Li-Yorke sensitivity of f¯. It will be shown that if f¯ is Li-Yoke sensitive, then f is Li-Yorke sensitive. Furthermore, we give an example showing that Li-Yorke sensitivity of f does not imply Li-Yorke sensitivity of f¯. This paper discusses the further work of [17]. And by suing the obtained results, we give positive answers to Sharma and Nagar’s question in [18].


## 2. The Denjoy Homeomorphism and an Interval Map

Let(X,d) be a compact metric space. For any nonempty subsets Y,Y′ of X and any r>0, write d(Y,Y′)=inf{d(x,y):x∈Y,y∈Y′}, diam(Y)=sup{d(x,y):x,y∈Y}, and B(Y,r)={x∈X:d(x,Y)<r}, where d(x,Y)=inf{d(x,y):y∈Y}. When Y={y} is a singleton, we write B(y,r) (resp., d(y,Y′)) for B(Y,r) (resp., d(Y,Y′)). For any nonempty subset  𝕂 of ℕ and any  i∈ℕ, write i+𝕂={i+n:n∈𝕂}.WriteN(U,V)={n∈ℕ:U∩f-n(V)≠Φ}, where  U,V are nonempty subsets in  X. A subset 𝕂⊂ℕ is syndetic (or relative dense) if there is N∈ℕ such that  {i,i+1,…,i+N}∩𝕂≠Φ for every i∈ℕ. A point x∈X is almost periodic if for any  ɛ>0, N(x,B(x,ɛ)) is syndetic. A subset 𝕂⊂ℕ is thick if it contains arbitrarily long runs of positive integers. A dynamical system is transitive if for each pair of nonempty open subsets  A,B of  X, N(A,B) is nonempty. A point  x∈X is transitive if the orbitO(x,f)≡{fn(x):n=0,1,2,…} is dense in X. A system  (X,f) is minimal if any x∈X is transitive. We say (X,f) is mixing if for each pair of nonempty open subsets U,V, N(U,V) is cofinite, and (X,f) is weakly mixing if (X×X,f×f) is transitive. The set ω(x,f)≡{y:thereexistsanincreasingsequence{ni}suchthaty=limi→∞fni(x)} is said to be the ω-limit set of x.Lemma 1.
If the system(X,f) is minimal, then for any x∈X and any open subset U⊂X, N(x,U) is syndetic. For some x∈X, if V⊂ω(x,f) is an invariant closed set with f(V)=V, then for any δ>0, N(x,B(V,δ)) is thick.Proof.
For anyx∈X and any open subset U⊂X there are δ>0 and n0∈ℕ such that fn0(B(x,δ))⊂U. It is well known that if the system (X,f) is minimal, then every v∈X is almost periodic. So N(x,B(x,δ)) is syndetic. Then N(x,U)⊃{i+n0:i∈N(B(x,δ)} is syndetic.
Sincef(V)=V and f is uniformly continuous, then for any δ>0 and any N∈ℕ, there is δ′∈(0,δ) such that
(3)fj(B(V,δ′))⊂B(V,δ),forj=1,…,2N.
So for some m∈ℕ with fm(x)∈B(V,δ′), {m,m+1,…,m+2N}⊂N(x,B(V,δ)).We will useℝ/ℤ as a model for the circle S1. The metric d′ is defined by d′(a,b)=min{|a-b|,1-|a-b|}. Rigid rotation by the real number α is then given by
(4)Rα:S1⟶S1,Rα=t+α(mod1).
Corresponding to the irrational α, the Denjoy homeomorphism dα:S1→S1 is an orientation preserving homeomorphism of the circle characterized by the following properties: the rotation number of dα is α; there is a Cantor set Cα⊂S1 on which dα acts minimally; and if u and v are any two components of S1∖Cα, then dαn(u)=v for some integer n (see [19]). There is a Cantor function hα:S1→S1 that semiconjugates dα with Rα:hα being a monotone surjection that collapses the components of S1∖Cα (and so maps Cα onto S1) with Rα∘hα=hα∘dα.Lemma 2.
Let(Cα,dα) be the minimal subsystem of a Denjoy homeomorphism, and c=max{diam(u):u is a connected component of S1∖Cα with diam(u)<1/4}. Then (Cα,dα) is c-sensitive. Furthermore, for any x∈Cα and any δ>0, there is y∈B(x,δ) such that Nc(x,y)≡{n:d′(dαn(x),(dαn(y)))>c} is syndetic.Proof.
For anyx∈Cα and any δ>0, there is y∈B(x,δ) such that hα(x)≠hα(y). Let [hα(x),hα(y)] be the arc in S1 whose endpoints are x and y and whose length is d′(hα(x),hα(y)). Then there exist w and δ′>0 such that w∈B(w,δ′)⊂[hα(x),hα(y)]. Let u be one of the connected components of S1∖Cα with diam(u)=c and p=dα(u). By Lemma 1, N(w,B(p,δ′)) is syndetic. For any i∈N(w,B(p,δ′)), p∈[Rαi(hα(x)),Rαi(hα(y))]. So d′(dαi(x),(dαi(y)))>c.Lemma 3.
Let(Cα,dα) be the minimal subsystem of a Denjoy homeomorphism Rα, and c=max{diam(u):u is a connected component of S1∖Cα with diam(u)<1/4}. Then for any x∈Cα, there is δ>0 such that for any y∈B(x,δ) with y≠x, liminfn→∞d′(dαn(x),dαn(y))>0.Proof.
Let{ui}i∈ℤ be an arrangement of the connected components of  S1∖Cα with dα(ui)=ui+1, i∈ℤ, and  diam(u0)=c. For any x∈S1, hα-1(x) has two elements at most. So for any v∈Cα, there is δ>0 such that for any y∈B(v,δ) with y≠v, hα(v)≠hα(y). For v′∈B(v,δ) and v′≠v, let [w,w′]=hα([v,v′]) be an arc, and p=hα(u0). For the irrational α, there exists k,l∈ℕ such that kαmod1<diam([w,w′]) and kα×lmod1<diam([w,w′]). So for any i∈ℕ, there is 0≤j≤l such that Rkαj(p)∈Rαi([w,w′]). So uk×j⊂dαi([v,v′]). Let ɛ0=min{diam(u0),diam(uk),…,diam(uk×l)}. Then liminfn→∞d′(dαn(v),dαn(v′))≥ɛ0>0.Lemma 4 (see [17]).
(
𝒦
(
C
α
)
,
d
¯
α
) is not sensitive (Cα is a stable point).Lemma 5 (see [6]).
If a nontrivial system(X,f) is weakly mixing then it is Li-Yorke sensitive.Lemma 6.
Letf:I→I be the tent map which is f(x)=1-|1-2x|. Then f is Li-Yorke sensitive.Proof.
It is well known that the tent map is mixing [12]. Apply Lemma 5.Example 7.
f
:
I
→
I is given by f|[0,1/3] and f|[2/3,1] which are the tent maps; f|[a,b] is a constant mapping, f|[1/3,a] and f|[b,2/3] are linear where 1/3<a<b<2/3 and f(a) is a transitive point of f|[0,1/3] (see Figure 1).Figure 1Lemma 8.
There is a positiveɛ>0, for any x∈(I∖⋃i=0∞‍f-i([a,b]))∪{2/3} and any δ>0, there exists y with d(x,y)<δ such that the pair  (x,y) is  ɛ-Li-Yorke sensitive.Proof.
By Lemma6, f|I∖(1/3,2/3) is Li-Yorke sensitive. Let  ɛ>0 be a Li-Yorke sensitive constant of f|I∖(1/3,2/3). Then for any x∈I∖(1/3,2/3), the lemma holds. For any  x∈(1/3,2/3)∖⋃i=0∞‍f-i([a,b]) and any  δ>0, there exist an open interval  U with  x∈U⊂B(x,δ)∩((1/3,2/3)∖⋃i=0∞‍f-i([a,b])) and n0∈ℕ such that fn0(U)⊂[0,1/3]. It is easy to see that  fn0(U) is connected open neighborhood of fn0(x). Because f|[0,1/3] is Li-Yorke sensitive, there is y∈U such that (x,y) is ɛ-Li-Yorke sensitive.


## 3. Li-Yorke Sensitivity

Lemma 9 (see [12]).
Let(X,f) be a system. Then the following statements are equivalent:(i)
fis weakly mixing;(ii)
f¯ is weakly mixing;(iii)
f¯ is transitive.Theorem 10.
If a nontrivial system(X,f) is weakly mixing, then f¯ is Li-Yorke sensitive.Proof.
By Lemma9, f¯ is weakly mixing. Apply Lemma 5.Theorem 11.
Iff¯ is Li-Yorke sensitive, then f is Li-Yorke sensitive.Proof.
Let(𝒦(X),f¯) be Li-Yorke sensitive. There exists  ɛ>0, for any {y}=Y∈𝒦(X) and any  δ>0, and there is a contract subset K with H(Y,K)<δ (so, for any  x∈K,  d(x,y)<δ) such that  (Y,K) is an  ɛ-Li-Yorke sensitive pair of  f¯. So there exists a point  y′∈K with  y′≠y such that (y,y′) is an ɛ-Li-Yorke sensitive pair of  f.


## 4. A Counter Example

Example 12.
Let(Cα,dα) be the minimal subsystem of a Denjoy homeomorphism, and c=max{diam(u):uisaconnectedcomponentofS1∖Cαwithdiam(u)<1/4}, and let (I,f) be the interval map given in Example 7. Let S={(r,2πθ):r∈I,θ∈Cα} be a subset in polar coordinate system with metric  ρ defined by
(5)ρ((r,θ),(r′,θ′))=(r2+r′2-2rr′cos(θ-θ′))1/2.
And let the map F:S→S be defined by F(r,2πθ)=(f(r),2πdα(θ)). It is easy to see that  (S,F) is a dynamical system.Proposition 13.
(
S
,
F
) is Li-Yorke sensitive.Proof.
For any(r,2πθ)∈S, either r∈(I∖⋃i=0∞‍f-i([a,b]))∪{2/3} or r∈⋃i=0∞‍f-i([a,b])∖{2/3}.
Ifr∈⋃i=0∞‍f-i([a,b])∖{2/3}, then there exists k∈ℕ such that fk(r)=f(a) is a transitive point of f|[0,1/3] and so ω(r,f)=[0,1/3]. Since 2/9 is a fixed point of f, by Lemma 1, N(r,B(2/9,1/9)) is thick. By Lemma 2, for any  δ>0, there exists  θ′∈B(x,δ/2π) such that  Nc(θ,θ′) is syndetic, so there is  m∈N(r,B(2/9,1/9))∩Nc(θ,θ′); that is, ρ(Fm(r,2πθ),Fm(r,2πθ′))=2r(1-cos2π(θ-θ′))1/2≥2/9(1-cos2πc)1/2. On the other hand, there is a sequence {ni}⊂ℕ such that limi→∞fni(r)=0. So limi→∞ρ(Fni(r,2πθ),Fni(r,2πθ′))=limi→∞2fni(r)(1-cos2π(θ-θ′))1/2=0. So ((r,2πθ), (r,2πθ′)) is a 2/9(1-cos2πc)1/2-Li-Yorke sensitive pair.
Ifr∈I∖⋃i=0∞‍f-i([a,b])∪{2/3}, by Lemma 8, there is a positive  ɛ, for any  δ>0, and there exists a point r′ with  d(r,r′)<δ such that (r,r′) is  ɛ-Li-Yorke sensitive. It is not difficult to verify that ((r,θ),(r′,θ)) is an ɛ-Li-Yorke sensitive.
To sum up,ɛ0=min{2/9(1-cos2πc)1/2,ɛ} is a Li-Yorke sensitive constant of F.Proposition 14.
(
𝒦
(
S
)
,
F
¯
) is not sensitive.Proof.
Writer0=(a+b)/2. By Lemma 4, Cα is a stable point of (𝒦(Cα),d¯α). So for any  ɛ>0, there exists 0<δ<(b-a)/2 such that for every  K∈𝒦(Cα) with Hd′(K,Cα)<δ and all i∈ℕ, Hd′(d¯αi(K),Cα)<ɛ.
We will prove that(r0,Cα) is a stable point of F¯. Let π1:S→I be the natural map defined by π1((r,θ))=r and  π2:S→Cα be the natural map defined by π2((r,θ))=θ. By the continuities of π1, π2, there exists 0<δ′<δ such that for any M∈𝒦(S) with  Hρ(M,(r0,Cα))<δ′, π1(M)⊂[a,b] and Hd′(π2(M),Cα)<δ. Then for any i∈ℕ,
(6)Hρ(F¯i(M),F¯i(r0,Cα))=Hρ(F¯i-1(f(a),d¯α(π2(M))),F¯i-1(f(a),Cα))≤Hd′(d¯αi(π2(M)),Cα)<ɛ.Proposition 15.
(
𝒦
(
S
)
,
F
¯
) is not spatiotemporal chaotic.Proof.
For any point(r0,2πθ), θ∈Cα. By Lemma 3, there is δ>0 such that for any θ′∈B(θ,δ) with θ′≠θ, liminfn→∞d′(dαn(θ),dαn(θ′))>0. By the continuities of π1,π2, there exists  0<δ′<δ such that for any M∈𝒦(S) with  Hρ(M,(r0,θ))<δ′, π1(M)⊂[a,b] and  Hd′(π2(M),θ)<δ. Let θ0∈B(θ,δ) with θ0≠θ. Then,
(7)liminfi→∞Hρ(F¯i(M),F¯i(r0,θ))=liminfi→∞Hρ(F¯i-1(f(a),d¯α(π2(M))),F¯i-1(f(a),dα(θ)))≤Hd′(d¯αi(π2(M)),Cα)<ɛ.From Propositions13 and 14 or from Propositions 13 and 15, we obtain the following at once.Theorem 16.
There is a dynamical system(X,f) such that (X,f) is Li-Yorke sensitive, but (𝒦(X),f¯) the set-valued discrete system induced by (X,f) is not sensitive.


## 5. Li-Yorke Sensitivity of Interval Maps

Lemma 17 (see [20]).
Letf:[a,b]→[a,b] be a transitive interval map. Then one of the following conditions holds:(i)
f is mixing;(ii)
there isc∈(a,b) such that if f([a,c])=[c,b] and f([c,b])=[a,c], in addition, c is the unique fixed point of f, and both f2|[a,c] and f2|[c,b] are mixing.Theorem 18.
Letf:[a,b]→[a,b] be a transitive interval map. Then f is Li-Yorke sensitive.Proof.
By Lemma17, either f is mixing or there is the unique fixed point c such that f2|[a,c] and f2|[c,b] are mixing.
Iff is mixing, then f is weakly mixing. Apply Lemma 5.
If there is the unique fixed pointc such that f2|[a,c] and f2|[c,b] are mixing, by Lemma 5, then f2|[a,c] and f2|[c,b] are Li-Yorke sensitive. It is easy to see that f is Li-Yorke sensitive.Example 19.
f
:
I
→
I is given by f|[0,1/3] and f|[2/3,1] which are the tent maps; f|[1/3,2/3] is linear. It is easy to see that f is Li-Yorke sensitive but is not transitive. So the converse version of Theorem 18 does not hold.Example 20.
f
:
I
→
I is given by f|[0,1/2] which is the tent map and f|[1/2,1] which is linear. It is not difficult to get that f is sensitive but is not Li-Yorke sensitive (1 is a distal point).The following example is an interval map which is spatiotemporal chaotic but is not Li-Yorke sensitive.Example 21.
f
:
I
→
I is given by f|[0,1/6], f|Ii and f|[2/3,1] which are the tent maps, and f|[1/6,1/3],  f|Ii′ are linear, where Ii=[(2/3)(1-(1/2)i),(2/3)(1-(1/2)i)+(1/2)i+1(1/3)], Ii′=[(2/3)(1-(1/2)i)+(1/2)i+1(1/3),(2/3)(1-(1/2)i+1)], i=1,2,… (see Figure 2).
For anyx∈I and any δ>0, there is n,n′∈ℕ such that fn′(x)∈In. Since f|In is mixing, there is y∈B(x,δ) such that x,y is proximal but is not asymptotic. So f is spatiotemporal chaotic.
On the other hand, for anyɛ>0, there exists n0∈ℕ such that diamIi<ɛ for all n>n0. Since f(Ii)=Ii, for all i∈ℕ, then any x∈⋃i=n0+3∞‍Ii is not ɛ-unstable (i.e., there exists  δ>0 such that diam(fi(B(x,δ)))<ɛ, for all i∈ℕ), so f is not sensitive; especially, f is not Li-Yorke sensitive.Figure 2



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