Some Chaotic Properties of Average Shadowing Property

Let be a metric space and be a continuous map. The notion of the -average shadowing property ( ASP ) for a continuous map on – space is introduced and the relation between the ASP and average shadowing property(ASP)is investigated. We show that if has ASP, then has ASP for every . We prove that if a map be pseudo-equivariant with dense set of periodic points and has the ASP, then is weakly mixing. We also show that if is a –expansive pseudo-equivariant homeomorphism that has the ASP and is topologically mixing, then has a -specification. We obtained that the identity map on has the ASP if and only if the orbit space ⁄ of is totally disconnected. Finally, we show that if is a pseudoequivariant map, and the trajectory map ⁄ is a covering map, then has the ASP if and only if the induced map  ̆ ⁄ ⁄ has ASP.


Introduction
The concept of shadowing property is one of the influential notions in the theory of dynamical systems. In 1967 The shadowing property (SP) was introduced by Anosov [1] and the concept of average shadowing property ( ASP ) was introduced by Blank for investigating chaotic dynamical systems [2]. In 1960, the notion of space was introduced by R. S. Palais [3]. The pseudo-

ISSN: 0067-2904
AL-Juboory and AL-Shara'a shadowing relies on the action of a group acting on . Also, she studied shadowing for the shift map on the contrary limit space produced by the map [5]. In section 1 of this paper., we study the ASP for continuous maps on spaces ( ASP). In section 2, we prove some similar results on the ASP in the metric space with some chaotic properties and we put sufficient conditions to prove these results on spaces.

Preliminaries
Let . Clearly, every equivariant map is a pseudo-equivariant map but the converse needs not to be true [6]. We introduce the definitions that we will need in this paper and recall some fundamental definitions. In this paper, we denote the metric space, on which there is a topological group with metric by by the map we mean . By being a compact metric space, we mean a compact metric space on which there is a compact topological group with metric . and . Thus, has the -specification by Definition 3.5.

Lemma 3.7. [5]
Let be a compact connected Hausdorff metric space that contains more than one point and let . Then for a continuous map and , there exists a pseudo-trajectory for containing in . We recall that the topological space is called a totally disconnected space if . There are two sets that are disconnection such that and .

Theorem 3.8
Let be a compact metric space. Then the identity map has the ASP if and only if the orbit space of is totally disconnected. Proof: Assume that the identity map has the ASP. By hypothesis, is compact, then it is enough to prove that . Suppose, conversely, that . Since so there is a closed connected subset Ε in which has a dimension that is at least one. is a compact subset of , since is compact. So , such that . By compactness of there is and such that . Let . We get a contradiction by exhibiting that for there is a average pseudo-trajectory for which is not ‫ـ‬ shadowed in average by the trajectory of some point . By Lemma 3.7, there is a average pseudo-trajectory for in containing . Such a average pseudo-trajectory can be obtained as follows: Since is a compact connected subset of by Lemma 3.7, then there is a pseudo-trajectory for ̌ containing and . This implies that , Since is Compact, implies for such that ,

∑ ( )
Moreover, using uniform continuity of the covering map we get : This proves that is ‫ـ‬ shadowed in average by . Hence, ̆ has the ASP.

Proof:
Assume that ̆ has the ASP. We must prove that has the ASP. We choose .