A Note on Specification Property of Dynamical Systems

: The paper is discussed the sensitive and transitive property of a dynamical system with strong specification property. It is proved that if a dynamical system is sensitive, then it is syndetically sensitive with the same constant of sensitivity. Further, it is given another condition such that if a dynamical system is sensitive, then it is syndetically sensitive with the same constant of sensitivity. Meanwhile, it is stated that if a dynamical system has shadowing property, then it is totally syndetically transitive.


Introduction
Specification property was first introduced by Bowen to give the distribution of periodic points for Axiom A diffeomorphisms [1]. In recent years, many scholars focus on specification property and its related properties. Lampart [2] described some relations between specification property and ω-chaos. Kulczycki [3] studied relations between almost specification property, asymptotic average shadowing property and average shadowing property for compact dynamical systems. Wang at. el. [4] shown that an expansive system with specification property displays a stronger form of ω-chaos. It is proved that there exists a system having almost specification property and zero entropy [5]. Kwietniak [6] examined relations between specification like properties and such notions as: mixing, entropy, the structure of the simplex of invariant measures, and various types of the shadowing property. Kwietniak at. el. [7] stated that the weak specification property neither implies intrinsic ergodicity, nor follows from almost specification. It was proved that a self homeomorphism on property [8].
The paper is organized in the following manner: In Section 2, it is given preliminaries required for the development of the paper. In Section 3, it is proved that if a dynamical system ( , ) X T is sensitive with strong specification property, then it is syndetically sensitive with the same constant of sensitivity.
Further, it is given another condition such that if ( , ) X T is sensitive, then it is syndetically sensitive with the same constant of sensitivity. At least, it is shown that if a dynamical system has d−shadowing property or d−shadowing property with strong specification property, then it is totally syndetically transitive.

Basic Definitions
Let : T X X → be a continuous map acting on a compact metric space ( , ) X d .  , , , k x x x ⋯ in X , any integers where # Λ denotes the cardinality of Λ . For a finite set of indices and the Bowen ball of radius ε centered at x by When ( , ) g n n ε < , it is defined the ( ; , ) g n ε Bowen ball X T has almost specification property with mistake function g , if for any 1 , and integers Remark 1. Pfister and Sullivan [9] showed that the strong specification property implies the almost specification property.
For any A ⊂ ℕ, the upper density of A is defined by Replacing limsup with liminf in (1) gives the definition of ( ) d A , the lower density of A . If there exists a number In the special case of X T has the ergodic shadowing property (resp., d−shadowing property and d−shadowing property). X T has a dense set of minimal points, the product system (

Specification Property of Dynamical Systems
also has a dense set of minimal points [10]. Therefore it can be chosen , T N U δ is also syndetic. Theorem 3.3. Let T has almost specification property and has an invariant measure with full support. If T is sensitive, then T is syndetically sensitive with the same constant of sensitivity. The following lemma follows from the ergodic theorem. Applying the proof process of Theorem 3.1, the Theorem 3.3 is proved.
Theorem 3.6. Let ( , ) X T be a dynamical system with strong specification property. If ( , ) X T has the d− shadowing property or d− shadowing property, then ( , ) X T is totally syndetically transitive. Lemma 3.7. If ( , ) X T is a topologically transitive system with a dense set of minimal points, then ( , ) X T is totally syndetically transitive.
Proof. Let , U V be non-empty open subsets of X containing points 1 y and 2 y , respectively. Choose a symmetric neighborhood U of the diagonal X ∆ such that Choose a positive integer ( ) M U as in the definition of TSP. For any sequence, , Thus, every open set V of X contains a periodic point. Hence, the set of periodic points of T is dense in X . Lemma 3.8 ([13]). For a dynamical system ( , ) X T , the following statements are equivalent: (1) T has the d−shadowing property (resp., d−shadowing property); (2) k T has the d−shadowing property (resp., d−shadowing property) for any k N ∈ ; (3) k T has the d−shadowing property (resp., d−shadowing property) for some k N ∈ .
Proof of Theorem 3.6. It is well known that every periodic points are minimal points.
By Lemma 3.2 ( , ) X T has a dense of minimal points. First, it is presented a proof for the case of the d− shadowing property. Given any pair of nonempty open subsets ,  Choose a sequence In particular, This proves that ( , ) X T is transitive, hence syndetically transitive by Lemma 3.7.

Conclusion
In the research of dynamical system, because the exact solutions of most systems cannot be obtained, scholars often use numerical calculation. The existence of computational errors inevitably leads to the production of "pseudo-orbit". It is well known that if a system has a "pseudo-orbit-shadowing property in the usual sense", then any pseudo-orbit with a sufficiently small single step error must be tracked by a real orbit-shadowing, and its "shadowing error"is uniformly bounded. This indicates that the numerically calculated pseudo-orbit can truly reflect the local dynamic behavior of the system in a certain sense.
In this paper, it is introduced several types of strong sensitivity and strong transitivity, and discussed the relationship between these properties in systems with different shadowing properties. It will lay a theoretical foundation for further exploring the dynamical behavior of the system.