Li-Yorke Sensitivity of Set-Valued Discrete Systems

Abstract

Consider the surjective, continuous map 𝑓∶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.

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 𝜀-sensitive if there exists a positive 𝜀 such that any 𝑥∈𝑋 is a limit of points 𝑦∈𝑋 satisfying the condition 𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))>𝜀 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, liminf_𝑛→∞𝑑𝑓^𝑛(𝑥),𝑓^𝑛𝑦=0,limsup_𝑛→∞𝑑𝑓^𝑛(𝑥),𝑓^𝑛𝑦>0. (1)

Li-Yoke sensitivity is introduced by Akin and Kolyada in [6]. A system is Li-Yorke sensitive if there exists 𝜀>0 such that every 𝑥∈𝑋 is a limit of points 𝑦∈𝑋 such that the pair (𝑥,𝑦) is proximal but sup_𝑛>𝑁{𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))}>𝜀 for any 𝑁>0, and the positive 𝜀 is said to be a Li-Yorke sensitive constant of the system. A pair (𝑥,𝑦) is 𝜀-Li-Yorke sensitive if the pair (𝑥,𝑦) is proximal but whose orbits are frequently at least 𝜀 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: 𝐻_𝑑𝐾₁,𝐾₂=maxsup𝑑𝑥₁,𝐾₂∶𝑥₁∈𝐾₁,sup𝑑𝑥₂,𝐾₂∶𝑥₂∈𝐾₂, (2) 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 of 𝑓 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}.

Write 𝑁(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 𝜀>0, 𝑁(𝑥,𝐵(𝑥,𝜀)) 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)≡{𝑓^𝑛(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{𝑛_𝑖}suchthat𝑦=lim_𝑖→∞𝑓^𝑛𝑖(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 any 𝑥∈𝑋 and any open subset 𝑈⊂𝑋 there are 𝛿>0 and 𝑛₀∈ℕ such that 𝑓^𝑛0(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.
Since 𝑓(𝑉)=𝑉 and 𝑓 is uniformly continuous, then for any 𝛿>0 and any 𝑁∈ℕ, there is 𝛿^′∈(0,𝛿) such that 𝑓^𝑗𝐵𝑉,𝛿^′⊂𝐵(𝑉,𝛿),for𝑗=1,…,2𝑁. (3) So for some 𝑚∈ℕ with 𝑓^𝑚(𝑥)∈𝐵(𝑉,𝛿^′), {𝑚,𝑚+1,…,𝑚+2𝑁}⊂𝑁(𝑥,𝐵(𝑉,𝛿)).

We will use ℝ/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 𝑅_𝛼∶𝑆¹⟶𝑆¹,𝑅_𝛼=𝑡+𝛼(mod1). (4) Corresponding to the irrational 𝛼, the Denjoy homeomorphism 𝑑_𝛼∶𝑆¹→𝑆¹ is an orientation preserving homeomorphism of the circle characterized by the following properties: the rotation number of 𝑑_𝛼 is 𝛼; there is a Cantor set 𝐶_𝛼⊂𝑆¹ on which 𝑑_𝛼 acts minimally; and if 𝑢 and 𝑣 are any two components of 𝑆¹⧵𝐶_𝛼, then 𝑑^𝑛_𝛼(𝑢)=𝑣 for some integer 𝑛 (see [19]). There is a Cantor function ℎ_𝛼∶𝑆¹→𝑆¹ that semiconjugates 𝑑_𝛼 with 𝑅_𝛼∶ℎ_𝛼 being a monotone surjection that collapses the components of 𝑆¹⧵𝐶_𝛼 (and so maps 𝐶_𝛼 onto 𝑆¹) with 𝑅_𝛼∘ℎ_𝛼=ℎ_𝛼∘𝑑_𝛼.

Lemma 2. Let (𝐶_𝛼,𝑑_𝛼) be the minimal subsystem of a Denjoy homeomorphism, and 𝑐=max{diam(𝑢)∶𝑢 is a connected component of 𝑆¹⧵𝐶_𝛼 with diam(𝑢)<1/4}. Then (𝐶_𝛼,𝑑_𝛼) is 𝑐-sensitive. Furthermore, for any 𝑥∈𝐶_𝛼 and any 𝛿>0, there is 𝑦∈𝐵(𝑥,𝛿) such that 𝑁_𝑐(𝑥,𝑦)≡{𝑛∶𝑑^′(𝑑^𝑛_𝛼(𝑥),(𝑑^𝑛_𝛼(𝑦)))>𝑐} is syndetic.

Proof. For any 𝑥∈𝐶_𝛼 and any 𝛿>0, there is 𝑦∈𝐵(𝑥,𝛿) such that ℎ_𝛼(𝑥)≠ℎ_𝛼(𝑦). Let [ℎ_𝛼(𝑥),ℎ_𝛼(𝑦)] be the arc in 𝑆¹ whose endpoints are 𝑥 and 𝑦 and whose length is 𝑑^′(ℎ_𝛼(𝑥),ℎ_𝛼(𝑦)). Then there exist 𝑤 and 𝛿^′>0 such that 𝑤∈𝐵(𝑤,𝛿^′)⊂[ℎ_𝛼(𝑥),ℎ_𝛼(𝑦)]. Let 𝑢 be one of the connected components of 𝑆¹⧵𝐶_𝛼 with diam(𝑢)=𝑐 and 𝑝=𝑑_𝛼(𝑢). By Lemma 1, 𝑁(𝑤,𝐵(𝑝,𝛿^′)) is syndetic. For any 𝑖∈𝑁(𝑤,𝐵(𝑝,𝛿^′)), 𝑝∈[𝑅^𝑖_𝛼(ℎ_𝛼(𝑥)),𝑅^𝑖_𝛼(ℎ_𝛼(𝑦))]. So 𝑑^′(𝑑^𝑖_𝛼(𝑥),(𝑑^𝑖_𝛼(𝑦)))>𝑐.

Lemma 3. Let (𝐶_𝛼,𝑑_𝛼) be the minimal subsystem of a Denjoy homeomorphism 𝑅_𝛼, and 𝑐=max{diam(𝑢)∶𝑢 is a connected component of 𝑆¹⧵𝐶_𝛼 with diam(𝑢)<1/4}. Then for any 𝑥∈𝐶_𝛼, there is 𝛿>0 such that for any 𝑦∈𝐵(𝑥,𝛿) with 𝑦≠𝑥, liminf_𝑛→∞𝑑^′(𝑑^𝑛_𝛼(𝑥),𝑑^𝑛_𝛼(𝑦))>0.

Proof. Let {𝑢_𝑖}_𝑖∈ℤ be an arrangement of the connected components of 𝑆¹⧵𝐶_𝛼 with 𝑑_𝛼(𝑢_𝑖)=𝑢_𝑖+1, 𝑖∈ℤ, and diam(𝑢₀)=𝑐. For any 𝑥∈𝑆¹, ℎ_𝛼⁻¹(𝑥) has two elements at most. So for any 𝑣∈𝐶_𝛼, there is 𝛿>0 such that for any 𝑦∈𝐵(𝑣,𝛿) with 𝑦≠𝑣, ℎ_𝛼(𝑣)≠ℎ_𝛼(𝑦). For 𝑣^′∈𝐵(𝑣,𝛿) and 𝑣^′≠𝑣, let [𝑤,𝑤^′]=ℎ_𝛼([𝑣,𝑣^′]) be an arc, and 𝑝=ℎ_𝛼(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 𝑅^𝑗_𝑘𝛼(𝑝)∈𝑅^𝑖_𝛼([𝑤,𝑤^′]). So 𝑢_𝑘×𝑗⊂𝑑^𝑖_𝛼([𝑣,𝑣^′]). Let 𝜀₀=min{diam(𝑢₀),diam(𝑢_𝑘),…,diam(𝑢_𝑘×𝑙)}. Then liminf_𝑛→∞𝑑^′(𝑑^𝑛_𝛼(𝑣),𝑑^𝑛_𝛼(𝑣^′))≥𝜀₀>0.

Lemma 4 (see [17]). (𝒦(𝐶_𝛼),𝑑_𝛼) is not sensitive (𝐶_𝛼 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. Let 𝑓∶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 1 

Lemma 8. There is a positive 𝜀>0, for any 𝑥∈(𝐼⧵⋃^∞_𝑖=0𝑓^−𝑖([𝑎,𝑏]))∪{2/3} and any 𝛿>0, there exists 𝑦 with 𝑑(𝑥,𝑦)<𝛿 such that the pair (𝑥,𝑦) is 𝜀-Li-Yorke sensitive.

Proof. By Lemma 6, 𝑓|_{𝐼⧵(1/3,2/3)} is Li-Yorke sensitive. Let 𝜀>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=0𝑓^−𝑖([𝑎,𝑏]) and any 𝛿>0, there exist an open interval 𝑈 with 𝑥∈𝑈⊂𝐵(𝑥,𝛿)∩((1/3,2/3)⧵⋃^∞_𝑖=0𝑓^−𝑖([𝑎,𝑏])) and 𝑛₀∈ℕ such that 𝑓^𝑛0(𝑈)⊂[0,1/3]. It is easy to see that 𝑓^𝑛0(𝑈) is connected open neighborhood of 𝑓^𝑛0(𝑥). Because 𝑓|_[0,1/3] is Li-Yorke sensitive, there is 𝑦∈𝑈 such that (𝑥,𝑦) is 𝜀-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 Lemma 9, 𝑓 is weakly mixing. Apply Lemma 5.

Theorem 11. If 𝑓 is Li-Yorke sensitive, then 𝑓 is Li-Yorke sensitive.

Proof. Let (𝒦(𝑋),𝑓) be Li-Yorke sensitive. There exists 𝜀>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 𝜀-Li-Yorke sensitive pair of 𝑓. So there exists a point 𝑦^′∈𝐾 with 𝑦^′≠𝑦 such that (𝑦,𝑦^′) is an 𝜀-Li-Yorke sensitive pair of 𝑓.

4. A Counter Example

Example 12. Let (𝐶_𝛼,𝑑_𝛼) be the minimal subsystem of a Denjoy homeomorphism, and 𝑐=max{diam(𝑢)∶𝑢isaconnectedcomponentof𝑆¹⧵𝐶_𝛼withdiam(𝑢)<1/4}, and let (𝐼,𝑓) be the interval map given in Example 7. Let 𝑆={(𝑟,2𝜋𝜃)∶𝑟∈𝐼,𝜃∈𝐶_𝛼} be a subset in polar coordinate system with metric 𝜌 defined by 𝜌(𝑟,𝜃),𝑟^′,𝜃^′=𝑟²+𝑟^′2−2𝑟𝑟^′cos𝜃−𝜃^′1/2. (5) And let the map 𝐹∶𝑆→𝑆 be defined by 𝐹(𝑟,2𝜋𝜃)=(𝑓(𝑟),2𝜋𝑑_𝛼(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=0𝑓^−𝑖([𝑎,𝑏]))∪{2/3} or 𝑟∈⋃^∞_𝑖=0𝑓^−𝑖([𝑎,𝑏])⧵{2/3}.
If 𝑟∈⋃^∞_𝑖=0𝑓^−𝑖([𝑎,𝑏])⧵{2/3}, then there exists 𝑘∈ℕ such that 𝑓^𝑘(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 𝑁_𝑐(𝜃,𝜃^′) is syndetic, so there is 𝑚∈𝑁(𝑟,𝐵(2/9,1/9))∩𝑁_𝑐(𝜃,𝜃^′); that is, 𝜌(𝐹^𝑚(𝑟,2𝜋𝜃),𝐹^𝑚(𝑟,2𝜋𝜃^′))=√2𝑟(1−cos2𝜋(𝜃−𝜃^′))^1/2≥√2/9(1−cos2𝜋𝑐)^1/2. On the other hand, there is a sequence {𝑛_𝑖}⊂ℕ such that lim_𝑖→∞𝑓^𝑛𝑖(𝑟)=0. So lim_𝑖→∞𝜌(𝐹^𝑛𝑖(𝑟,2𝜋𝜃),𝐹^𝑛𝑖(𝑟,2𝜋𝜃^′))=lim_𝑖→∞√2𝑓^𝑛𝑖(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.
If 𝑟∈𝐼⧵⋃^∞_𝑖=0𝑓^−𝑖([𝑎,𝑏])∪{2/3}, by Lemma 8, there is a positive 𝜀, for any 𝛿>0, and there exists a point 𝑟^′ with 𝑑(𝑟,𝑟^′)<𝛿 such that (𝑟,𝑟^′) is 𝜀-Li-Yorke sensitive. It is not difficult to verify that ((𝑟,𝜃),(𝑟^′,𝜃)) is an 𝜀-Li-Yorke sensitive.
To sum up, 𝜀₀=min{√2/9(1−cos2𝜋𝑐)^1/2,𝜀} is a Li-Yorke sensitive constant of 𝐹.

Proposition 14. (𝒦(𝑆),𝐹) is not sensitive.

Proof. Write 𝑟₀=(𝑎+𝑏)/2. By Lemma 4, 𝐶_𝛼 is a stable point of (𝒦(𝐶_𝛼),𝑑_𝛼). So for any 𝜀>0, there exists 0<𝛿<(𝑏−𝑎)/2 such that for every 𝐾∈𝒦(𝐶_𝛼) with 𝐻_𝑑′(𝐾,𝐶_𝛼)<𝛿 and all 𝑖∈ℕ, 𝐻_𝑑′(𝑑^𝑖_𝛼(𝐾),𝐶_𝛼)<𝜀.
We will prove that (𝑟₀,𝐶_𝛼) is a stable point of 𝐹. Let 𝜋₁∶𝑆→𝐼 be the natural map defined by 𝜋₁((𝑟,𝜃))=𝑟 and 𝜋₂∶𝑆→𝐶_𝛼 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 𝐻_𝜌(𝑀,(𝑟₀,𝐶_𝛼))<𝛿^′, 𝜋₁(𝑀)⊂[𝑎,𝑏] and 𝐻_𝑑′(𝜋₂(𝑀),𝐶_𝛼)<𝛿. Then for any 𝑖∈ℕ, 𝐻_𝜌𝐹^𝑖(𝑀),𝐹^𝑖𝑟₀,𝐶_𝛼=𝐻_𝜌𝐹^𝑖−1𝑓(𝑎),𝑑_𝛼𝜋₂(𝑀),𝐹^𝑖−1𝑓(𝑎),𝐶_𝛼≤𝐻_𝑑′𝑑^𝑖_𝛼𝜋₂(𝑀),𝐶_𝛼<𝜀. (6)

Proposition 15. (𝒦(𝑆),𝐹) is not spatiotemporal chaotic.

Proof. For any point (𝑟₀,2𝜋𝜃), 𝜃∈𝐶_𝛼. By Lemma 3, there is 𝛿>0 such that for any 𝜃^′∈𝐵(𝜃,𝛿) with 𝜃^′≠𝜃, liminf_𝑛→∞𝑑^′(𝑑^𝑛_𝛼(𝜃),𝑑^𝑛_𝛼(𝜃^′))>0. By the continuities of 𝜋₁,𝜋₂, there exists 0<𝛿^′<𝛿 such that for any 𝑀∈𝒦(𝑆) with 𝐻_𝜌(𝑀,(𝑟₀,𝜃))<𝛿^′, 𝜋₁(𝑀)⊂[𝑎,𝑏] and 𝐻_𝑑′(𝜋₂(𝑀),𝜃)<𝛿. Let 𝜃₀∈𝐵(𝜃,𝛿) with 𝜃₀≠𝜃. Then, liminf_𝑖→∞𝐻_𝜌𝐹^𝑖(𝑀),𝐹^𝑖𝑟₀,𝜃=liminf_𝑖→∞𝐻_𝜌𝐹^𝑖−1𝑓(𝑎),𝑑_𝛼𝜋₂(𝑀),𝐹^𝑖−1𝑓(𝑎),𝑑_𝛼(𝜃)≤𝐻_𝑑′𝑑^𝑖_𝛼𝜋₂(𝑀),𝐶_𝛼<𝜀. (7)

From Propositions 13 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]). Let 𝑓∶[𝑎,𝑏]→[𝑎,𝑏] be a transitive interval map. Then one of the following conditions holds:(i)𝑓 is mixing;(ii)there is 𝑐∈(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. Let 𝑓∶[𝑎,𝑏]→[𝑎,𝑏] be a transitive interval map. Then 𝑓 is Li-Yorke sensitive.

Proof. By Lemma 17, either 𝑓 is mixing or there is the unique fixed point 𝑐 such that 𝑓²|_[𝑎,𝑐] and 𝑓²|_[𝑐,𝑏] are mixing.
If 𝑓 is mixing, then 𝑓 is weakly mixing. Apply Lemma 5.
If there is the unique fixed point 𝑐 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|_𝐼′𝑖 are linear, where 𝐼_𝑖=[(2/3)(1−(1/2)^𝑖),(2/3)(1−(1/2)^𝑖)+(1/2)^𝑖+1(1/3)], 𝐼^′_𝑖=[(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 any 𝑥∈𝐼 and any 𝛿>0, there is 𝑛,𝑛^′∈ℕ such that 𝑓^𝑛′(𝑥)∈𝐼_𝑛. 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 any 𝜀>0, there exists 𝑛₀∈ℕ such that diam𝐼_𝑖<𝜀 for all 𝑛>𝑛₀. Since 𝑓(𝐼_𝑖)=𝐼_𝑖, for all 𝑖∈ℕ, then any 𝑥∈⋃^∞_𝑖=𝑛0+3𝐼_𝑖 is not 𝜀-unstable (i.e., there exists 𝛿>0 such that diam(𝑓^𝑖(𝐵(𝑥,𝛿)))<𝜀, for all 𝑖∈ℕ), so 𝑓 is not sensitive; especially, 𝑓 is not Li-Yorke sensitive.

Figure 2 

Acknowledgments

This work is supported by NSFC 11001038 and the Fundamental Research Funds for the Central Universities DC120101112.