On local aspects of sensitivity in topological dynamics

In this paper along with the research on weakly mixing sets and transitive sets, we introduce a local aspect of sensitivity in topological dynamics and give the concept of an s-set. It is shown that a weakly mixing set is an s-set. A transitive set with the set of periodic points being dense is an s-set. In particular, a transitive set is an s-set for interval maps. Moreover, we discuss s-sets for set-valued discrete dynamical systems.


Introduction
A topological dynamical system (abbreviated by TDS) is a pair (X, f ), where X is a compact metric space with metric d and f : X → X is a continuous map. When X is finite, it is a discrete space and there is no non-trivial convergence at all. Hence, we assume that X contains infinitely many points. Let N + denote the set of all positive integers and let N = N + ∪ {}.
Transitivity, weak mixing, and sensitive dependence on initial conditions (see [-]) are global characteristics of topological dynamical systems. Let (X, f ) be a TDS, (X, f ) is (topologically) transitive if for any nonempty open subsets U and V of X there exists an n ∈ N such that f n (U) ∩ V = ∅. (X, f ) is (topologically) mixing if for any nonempty open subsets U and V of X, there exists an N ∈ N such that f n (U) ∩ V = ∅ for all n ∈ N with n ≥ N . (X, f ) is (topologically) weakly mixing if for any nonempty open subsets U  , U  , V  , and V  of X, there exists an n ∈ N such that f n (U  ) ∩ V  = ∅ and f n (U  ) ∩ V  = ∅. It follows from these definitions that mixing implies weak mixing, which in turn implies transitivity. A map f is said to have sensitive dependence on initial conditions if there is a constant δ >  such that for any nonempty open set U of X, there exist points x, y ∈ U such that d(f n (x), f n (y)) > δ for n ∈ N + .
In [], Blanchard introduced overall properties and partial properties. For example, sensitive dependence on initial conditions, Devaney chaos (see []), weak mixing, mixing and more belong to overall properties; Li-Yorke chaos (see []) and positive entropy (see [, ]) belong to partial properties. Weak mixing is an overall property, it is stable under semiconjugate maps and implies Li-Yorke chaos. We find that a weakly mixing system always contains a dense uncountable scrambled set (see []). In [], Blanchard and Huang intro-duced the concepts of a weakly mixing set, derived from a result given by Xiong and Yang [] and showed 'partial weak mixing implies Li-Yorke chaos' and 'Li-Yorke chaos cannot imply partial weak mixing' .
Motivated by the idea of Blanchard and Huang's notion of a 'weakly mixing subset' , Oprocha and Zhang [] extended the notion of a weakly mixing set and gave the concept of a transitive set and discussed its basic properties. In this paper we give the concept of 's-set' for topological dynamical systems and investigate the relationship among transitive subsets, weakly mixing sets, and s-sets. We find that a TDS to have a weakly mixing set implies it has an s-set, and if periodic points are dense in the transitive set, then the transitive set is an s-set. In particular, a transitive set is an s-set for interval maps. The properties of transitivity, weak mixing, and sensitivity on initial conditions for a set-valued discrete dynamical system were discussed (see [-]). Also, we continue to discuss s-sets for setvalued discrete dynamical systems and investigate the relationship between a set-valued discrete system and an original system on an s-set. More precisely, a set-valued discrete system has an s-set, which implies that an original system has an s-set.

Preliminaries
A TDS (X, f ) is point transitive if there exists a point x  ∈ X with dense orbit i.e., orb(x  ) = X. Such a point x  is called a transitive point of (X, f ). In general, transitive and point A map f is said to be Devaney chaotic if f satisfies the following conditions: () f is transitive, () f is periodically dense; i.e., the set of periodic points of f is dense in X, and () f is sensitive dependent on initial conditions.

Definition . []
Let (X, f ) be a TDS and A be a closed subset of X with at least two elements. A is said to be weakly mixing if for any k ∈ N, any choice of nonempty open According to the definitions of transitive set and weakly mixing subset, we have the following results.
Result . If A is a weakly mixing set of (X, f ), then A is a transitive set of (X, f ).

Definition . A nonempty subset
A is called an s-set of (X, f ) if there exists a δ >  such that for any x ∈ A and ε > , there exist a y ∈ B(x, ε) ∩ A and an n ∈ N + satisfying d(f n (x), f n (y)) > δ.
Remark . The s-set is dense in itself, i.e., it contains no isolated points. X is an s-set of (X, f ) if and only if (X, f ) is sensitive dependent on initial conditions.
Let (X, f ) and (Y , g) be two TDSs.
If further h is a homeomorphism, then (X, f ) and (Y , g) are said to be topologically conjugate and the homeomorphism h is called a conjugated map.
Theorem . Let A be a weakly mixing set of (X, f ). Then A is an s-set of (X, f ).
Proof Let A be a weakly mixing set of (X, f ). Then A contains at least two points. Pick up two distinct points For any x ∈ A and ε > , we see that We consider an open subset B(x, ε) ∩ A of A and two open subsets B(x  , δ), B(x  , δ) of X. Since A is a weakly mixing set and Therefore, either d(f n (x), f n (y  )) > δ or d(f n (x), f n (y  )) > δ. This shows that A is an s-set of (X, f ).

Characterizing s-sets
In this section, we discuss the properties of s-sets of (X, f ). For a TDS (X, f ) and two nonempty subsets U, V ⊆ X, we use the following notation:

Lemma . Let A be an infinite subset of X and P(f ) be dense in A. Then P(f ) ∩ A is infinite.
Proof Suppose that P(f ) ∩ A is finite and let card(P(f ) ∩ A) = n, where card represents the cardinality of a set. Since A is an infinite subset of X, there exists a pairwise disjoint open set V A i of A for i = , , . . . , n + , i.e., V A i ∩ V A j = ∅ for i, j ∈ {, , . . . , n + } and i = j. Moreover, P(f ) ∩ A is dense in A, which implies card(P(f ) ∩ A) ≥ n + . This is a contradiction. Therefore, P(f ) ∩ A is infinite.

Theorem . Let (X, f ) be a TDS and A be an infinite subset of X. If A is a transitive set of (X, f ) and P(f ) is dense in A, then A is an s-set of (X, f ).
Proof We first prove that there exists δ  >  such that, for any x ∈ A, there exists q ∈ P(f ) ∩ A satisfying Indeed, by Lemma ., P(f ) ∩ A is an infinite set. Hence, we pick two different points Then, for any x ∈ A, we have Hence, by the triangle inequality, we have This is a contradiction by (.). Take δ = δ   . For any x ∈ A and ε > , without loss of generality, let ε < δ. Since P(f ) is dense in A, we have P(A) ∩ (B(x, ε) ∩ A) = ∅. Furthermore, we can take p ∈ B(x, ε) ∩ A and let f n (p) = p. Since x ∈ A, there exists q ∈ P(f ) ∩ A such that d orb(q), x ≥ δ.
Since f n (p) = p, we have f nj (p) = p. Hence, by the triangle inequality, d(p, x).
Again, by the triangle inequality, we have for some p ∈ B(x, ε) ∩ A and some y ∈ B(x, ε) ∩ A. Therefore, A is an s-set of (X, f ).
By [], if X is a non-degenerate compact interval, f : X → X is a continuous map and f is transitive, then P(f ) is dense in X. We prove that if X is a non-degenerate compact interval, A is a non-degenerate closed interval, and A is a transitive set of (X, f ), then P(f ) is dense in A.

Lemma . []
Suppose that I is a non-degenerate interval and f : I → I is a continuous map. If J ⊆ I is an interval which contains no periodic points of f and z, f m (z) and f n (z) ∈ J with  < m < n, then either z < f m (z) < f n (z) or z > f m (z) > f n (z). is an open interval of I and J  ⊆ J  and J is a transitive set, there exist n > m and z ∈ U such that f n (z) ∈ J  . Furthermore, we have  < m < n and z, f n (z) ∈ J  while f m (z) / ∈ J  . This is a contradiction by Lemma .. Therefore,

Example . We have the tent map (see Figures  and )
which is called Devaney chaos on I = [, ] by []. We will prove that [   ,   ] is a transitive set of (I, f ).
Let S(f k ) denote the set of extreme value points of f k for every k ∈ N + . Then For any nonempty open set U of [   ,   ], without loss of generality, we take U = (x ε, x  + ε) for a given ε >  and

s-sets for set-valued discrete dynamical systems
The distance from a point x to a nonempty set A in X is defined by Let κ(X) be the family of all nonempty compact subsets of X. The Hausdorff metric on κ(X) is defined by Thenf is well defined. (κ(X),f ) is called a set-valued discrete dynamical system. Let X be a T  space, that is, a single point set that is closed. Then κ(A) = {F ∈ κ(X) : F ⊆ A} is a closed subset of κ(X) for any nonempty closed subset A of X (see []).

Proof Let V κ(A) be a nonempty open subset of κ(A). Then there exists an open set
Since P(f ) is dense in A, it follows that P(f ) ∩ (V i ∩ A) = ∅ for each i = , , . . . , m. Furthermore, there exist y i ∈ P(f ) ∩ (V i ∩ A) and n i ∈ N + such that f n i (y i ) = y i for each i = , , . . . , m. Let G = {y  , y  , . . . , y m }. Then G ∈ V and G ∈ κ(A), which implies G ∈ V κ(A) . Moreover, f n  n  ···n m (y i ) = y i for each i = , , . . . , m. Therefore, (f ) n  n  ···n m (G) = f n  n  ···n m (G) = G, it means that P(f ) ∩ V κ(A) = ∅. This shows that P(f ) is dense in κ(A).

Theorem . Let A be a nonempty closed subset of X. If κ(A)
is a sensitive set of (κ(X),f ), then A is an s-set of (X, f ).
Let x ∈ A and ε > . Take  Since d H (f n ({x}), f n (G)) = sup y∈G d(f n (x), f n (y)), G is a compact subset of X and f : X → X is a continuous map, there exists y  ∈ G such that d H f n {x} , f n (G) = d f n (x), f n (y  ) > δ. G ⊆ B(x, ε) and G ⊆ A, consequently, y  ∈ B(x, ε) ∩ A. This shows that A is an s-set of (X, f ).