Li-Yorke Sensitivity of Set-Valued Discrete Systems

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 iff is Li-Yoke sensitive, thenf is Li-Yorke sensitive. Furthermore, we give an example showing that Li-Yorke sensitivity of f does not imply Li-Yorke sensitivity of f.


Introduction
Throughout this paper a dynamical system (, ) is a pair where  is a compact metric space with metric  and  :  →  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 (, ) is called -sensitive if there exists a positive  such that any  ∈  is a limit of points  ∈  satisfying the condition (  (),   ()) >  for some positive integer .According to Li and Yorke (see [5]), a subset  ⊂  is a scrambled set (for ), if any different points  and  from  are proximal and not asymptotic; that is, lim inf  → ∞  (  () ,   ()) = 0, lim sup  → ∞  (  () ,   ()) > 0. (1) Li-Yoke sensitivity is introduced by Akin and Kolyada in [6].A system is Li-Yorke sensitive if there exists  > 0 such that every  ∈  is a limit of points  ∈  such that the pair (, ) is proximal but sup > {(  (),   ())} >  for any  > 0, and the positive  is said to be a Li-Yorke sensitive constant of the system.A pair (, ) is -Li-Yorke sensitive if the pair (, ) is proximal but whose orbits are frequently at least  apart.
A dynamical system (, ) 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  ∈  and any open subset  with  ∈ , there is  ∈  such that  and  are proximal and not asymptotic.It is easy to see that Li-Yorke sensitivity implies spatiotemporal chaos and sensitivity.
In this paper, we discuss the relationship between Li-Yorke sensitivity of  and Li-Yorke sensitivity of .It will be shown that if  is Li-Yoke sensitive, then  is Li-Yorke sensitive.Furthermore, we give an example showing that there exists an increasing sequence {  } such that  = lim  → ∞    ()} is said to be the -limit set of .
We will use R/Z as a model for the circle  1 .The metric   is defined by   (, ) = min{| − |, 1 − | − |}.Rigid rotation by the real number  is then given by Corresponding to the irrational , the Denjoy homeomorphism   :  1 →  1 is an orientation preserving homeomorphism of the circle characterized by the following properties: the rotation number of   is ; there is a Cantor set   ⊂  1 on which   acts minimally; and if  and V are any two components of  1 \   , then    () = V for some integer  (see [19]).There is a Cantor function ℎ  :  Proof.Let {  } ∈Z be an arrangement of the connected components of  1 \   with   (  ) =  +1 ,  ∈ Z, and diam( 0 ) = .For any  ∈  1 , ℎ −1  () has two elements at most.So for any V ∈   , there is  > 0 such that for any  ∈ (V, ) with  ̸ = V, ℎ  (V) ̸ = ℎ  ().For V  ∈ (V, ) and ) be an arc, and  = ℎ  ( 0 ).For the irrational , there exists ,  ∈ N such that  mod 1 < diam([,   ]) and  ×  mod 1 < diam([,   ]).So for any  ∈ N, there is 0 ≤  ≤  such that Lemma 4 (see [17]).(K(  ),   ) is not sensitive (  is a stable point).
Lemma 6.Let  :  →  be the tent map which is Proof.It is well known that the tent map is mixing [12].Apply Lemma 5.

Li-Yorke Sensitivity of Interval Maps
Lemma 17 (see [20]).Let  : [, ] → [, ] be a transitive interval map.Then one of the following conditions holds:  Example 20.  :  →  is given by | [0,1/2] which is the tent map and | [1/2,1] which is linear.It is not difficult to get that  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.For any  ∈  and any  > 0, there is ,   ∈ N such that    () ∈   .Since |   is mixing, there is  ∈ (, ) such that ,  is proximal but is not asymptotic.So  is spatiotemporal chaotic.

Theorem 10 .Theorem 11 .
If a nontrivial system (, ) is weakly mixing, then  is Li-Yorke sensitive.Proof.By Lemma 9,  is weakly mixing.Apply Lemma 5.If  is Li-Yorke sensitive, then  is Li-Yorke sensitive.