The Research of Sequence Shadowing Property and Regularly Recurrent Point on the Double Inverse Limit Space
Abstract
According to the definition of sequence shadowing property and regularly recurrent point in the inverse limit space, we introduce the concept of sequence shadowing property and regularly recurrent point in the double inverse limit space and study their dynamical properties. The following results are obtained: (1) Regularly recurrent point sets of the double shift map σf∘σg are equal to the double inverse limit space of the double self-map f∘g in the regularly recurrent point sets. (2) The double self-map f∘g has sequence shadowing property if and only if the double shift map σf∘σg has sequence shadowing property. Thus, the conclusions of sequence shadowing property and regularly recurrent point are generalized to the double inverse limit space.
1. Introduction
Inverse limit space plays an important role in topological dynamical systems, and a series of research results have been obtained (see [1–4]). With the gradual deepening of problem research, it is found that the finite symbols in the inverse limit space have been difficult to solve the practical problems in life. Therefore, scholars began to try to extend the inverse limit space to the double inverse limit space and study the dynamical properties of Lipschitz tracking property, topological transitivity, and nonwandering points in the double inverse limit space. The relevant results are shown in [5–8]. Up to now, there are few new results in the double inverse limit space and it needs to be further studied. In addition, it is known that shadowing property and regularly recurrent point have always been the focus of topological dynamical systems. At present, many scholars have studied them in the inverse limit space and obtained valuable research results (see [9–15]). But there are few research results about them in the double inverse limit space. In this paper, firstly, the new concepts of sequence shadowing property and regularly recurrent point are given in double inverse limit space. Secondly, by using the method of mutual disclosure of the double self-map f∘g and the double shift map σf∘σg, we can obtain the following theorem.
Theorem 1. Let (X, d) be a compact metric space, f : X⟶X be a continuous map, g : X⟶X be a continuous map, and f∘g = g∘f. Then, we have that
Theorem 2. Let (X, d) be a compact metric space, f : X⟶X be a homeomorphic map, g : X⟶X be a homeomorphic map, and f∘g = g∘f. Then, the double self-map f∘g has sequence shadowing property if and only if the double shift map σf∘σg has sequence shadowing property.
The results enrich the conclusions of sequence shadowing property and regularly recurrent point in the double inverse limit space.
2. Regularly Recurrent Point in the Double Inverse Limit Space
Definition 3. Let (X, d) be a metric space and f be a continuous map from X to X. f is said to be a homeomorphic map if f is one-to-one and the map f and f−1 are continuous.
Definition 4 (see [7].)Let (X, d) be a compact metric space, f : X⟶X be a continuous map, g : X⟶X be a continuous map and f∘g = g∘f. Xf∘g is said to be the double inverse limit spaces of X if we write
where we write .
Then, the map πi,j is a continuous, surjective, and open map.
Definition 5. Let (X, d) be a metric space and f be a continuous map from X to X. A point x ∈ X is called to be a regularly recurrent point if each open set x ∈ U, there exists positive integer m > 0 such that for any positive integer k > 0, we have fkm(x) ∈ U, denoted by RR(f) the regularly recurrent point set of the map f.
Remark 6. According to the concept of regularly recurrent point of the map f, we will give the concept of regularly recurrent point of the double map f∘g.
Definition 7. Let (X, d) be a compact metric space, f : X⟶X be a continuous map, g : X⟶X be a continuous map, and f∘g = g∘f. A point x ∈ X is called to be a regularly recurrent point if each open set x ∈ U, there exists positive integer n, m > 0 such that for any positive integer k, l > 0, we have fkn∘glm(x) ∈ U, denoted by RR(f∘g) the regularly recurrent points set of the double map f∘g.
In order to prove the main theorem in this section, we will give two lemmas below.
Lemma 8 (see [8].)Let (X, d) be a compact metric space, f : X⟶X be a continuous map, g : X⟶X be a continuous map, and f∘g = g∘f. Then, for any positive integer i, j ∈ Z and nonnegative integer m, n ≥ 0, we have
Lemma 9. Let (X, d) be a compact metric space, f : X⟶X be a continuous map, g : X⟶X be a continuous map, and f∘g = g∘f. Then, we have that
Proof. Suppose x ∈ RR(f∘g). Let U be an any open set containing the point f∘g(x). Then, (f∘g)−1(U) is an open set containing the point x. According to x ∈ RR(f∘g), there exists positive integer n, m > 0 such that for any positive integer k, l > 0, we have
Thus, fkn∘glm(f∘g(x)) ∈ U. So f∘g(x) ∈ RR(f∘g). Hence,f∘g(RR(f∘g)) ⊂ RR(f∘g).
Remark 10. The regularly recurrent point set RR(f∘g) is invariant to the double map f∘g. So we can study the double inverse limit space of the double self-map f∘g in the regularly recurrent point set RR(f∘g). Now let us give the proof process of Theorem 11.
Theorem 11. Let (X, d) be a compact metric space, f : X⟶X be a continuous map, g : X⟶X be a continuous map, and f∘g = g∘f. Then, we have
Proof. Suppose . For any integer i, j ∈ Z, let U be an any open set containing the point xi,j. Thus, is an open set containing the point . According to , there exists positive integer n, m > 0 such that for any positive integer k, l > 0, we have that
Thus, we have that
By Lemma 8, we can get that
That is,
So xi,j ∈ RR(f∘g). Hence, .
Suppose . Then, for any integer i, j ∈ Z, we have
Let V be an any open set containing the point . Then, πi,j(V) is an open set containing the point yi,j. Thus, there exists positive integer t1, t2 > 0 such that for any positive integer p, s > 0, we have that
Thus, we have that
By Lemma 8, we can get that
Thus, we have that
So . Hence, . This completes the proof.
3. Sequence Shadowing Property in the Double Inverse Limit Space
Definition 12 (see [4].)Let (X, d) be a metric space and f be a continuous map from X to X. The sequence is called δ-pseudo orbit of f if for any i ≥ 0, we have d(f(xi), xi+1) < δ.
Definition 13 (see [4].)Let (X, d) be a metric space and f be a continuous map from X to X. The sequence is said to be ε-shadowed by the point x in X if for any i ≥ 0, we have d(fi(x), xi) < ε.
Definition 14 (see [4].)Let (X, d) be a metric space and f be a continuous map from X toX. The map f has a sequence shadowing property if each ε > 0, there exists δ > 0 such that any δ-pseudo orbit of f, there exists a point x in X and nonnegative integer sequence such that the sequence is ε-shadowed by the pointx.
Remark 15. By Definitions 12–14, we will give the concept of sequence shadowing property of the double map f∘g.
Definition 16. Let (X, d) be a compact metric space, f : X⟶X be a continuous map, g : X⟶X be a continuous map, and f∘g = g∘f. The sequence is called δ-pseudo orbit of the double map f∘g if for any integer i, j ∈ Z, we have d(f∘g(xi,j), xi+1,j+1) < δ.
Definition 17. Let (X, d) be a compact metric space, f : X⟶X be a continuous map, g : X⟶X be a continuous map, and f∘g = g∘f. The sequence is said to be ε-shadowed by the point x in X if for any positive integer i, j ∈ Z, we have
Definition 18. Let (X, d) be a compact metric space, f : X⟶X be a continuous map, g : X⟶X be a continuous map, and f∘g = g∘f. The double map f∘g has sequence shadowing property if each ε > 0, there exists δ > 0 such that for any δ-pseudo orbit of f∘g, there exists a point x in X and increasing positive integer sequence and such that the sequence is ε-shadowed by the point x ∈ X.
Now we begin to prove Theorem 19.
Theorem 19. Let (X, d) be a compact metric space, f : X⟶X be a homeomorphic map, g : X⟶X be a homeomorphic map, and f∘g = g∘f. Then, the double map f∘g has sequence shadowing property if and only if the double shift map σf∘σg has sequence shadowing property.
Proof. Suppose the double map f∘g has the sequence shadowing property. Since X is compact metric space, it is bounded. Write M = diam(X). Then, for any ε > 0, there exists positive integer m1 > 0 and n1 > 0 such that
Since the map f and g is uniformly continuous, it follows that for any 0 ≤ i ≤ 2m1 and 0 ≤ j ≤ 2n1, there exists 0 < δ1 < ε such that d(u, v) < δ1 implies
Note that the double map f∘g has the sequence shadowing property, and it follows that there exists 0 < δ2 < δ1 such that for any δ2-pseudo orbit of f∘g, there exists a point x ∈ X and increasing positive integer sequence and such that the sequence is δ1-shadowed by the point x. Let be -pseudo orbit of the double map σf∘σg where . Then, for any integer s, t ∈ Z, we have that
According to the definition of the metric , for any integer s, t ∈ Z, we can get that
So we have that
Thus, is δ2-pseudo orbit of the double map f∘g in X. Hence, there exists x ∈ X and increasing positive integer sequence and such that for any integer s, t ∈ Z, we have that
By (22), for any 0 ≤ i ≤ 2m1 and 0 ≤ j ≤ 2n1, we can get
Let and . For any i ≥ 0 and j ≥ 0, we have that
So the point is in Xf∘g. It is easy to know that
When |h| > m1 or |k| > n1, we can, respectively, get that
When |h| > m1 and |k| > n1, for any 0 ≤ i ≤ 2m1 and 0 ≤ j ≤ 2n1, we have that
where m1 − h = i and n1 − k = j.
By (27), we can obtain that
Thus, for any h, k ∈ Z we can get that
So for any integer s, t ∈ Z, we have that
Hence, the double shift map σf∘σg has the sequence shadowing property.
Suppose the double shift map has the sequence shadowing property. For each η > 0, there exists δ3 > 0 such that for any δ3-pseudo orbit of σf∘σg, there exists a point y ∈ X and increasing positive integer sequence and such that the sequence is η-shadowed by the point y. Let m2 > 0 and n2 > 0 such that
Since the map f and g is uniformly continuous, it follows that for any −m2 ≤ s ≤ m2 and−n2 ≤ t ≤ n2, there exists 0 < δ4 < δ3 such that d(u, v) < δ4 implies
Let be δ4-pseudo orbit of the double map f∘g. Then, for any i, j ∈ Z, we have that
By (36), when −m2 ≤ s ≤ m2 and −n2 ≤ t ≤ n2, we have that
For every yi,j, let . It is easy to know that
When |h| > m2 or |k| > n2, we can, respectively, get that
When |h| > m2 and |k| > n2, for any −m2 ≤ s ≤ m2 and −n2 ≤ t ≤ n2, we have that
where −h = s and −k = t.
By (38), we can get that
So for any integer h, k ∈ Z, we have that
Thus for any integer i, j ∈ Z, we have that
Hence, is δ3-pseudo orbit of the double map σf∘σg in Xf°g. So there exists a point and increasing positive integer sequence and such that for any integer i, j ∈ Z, we have that
According to the definition of the metric , we can get that
Hence, the map f∘g has the sequence shadowing property. Thus, we end the proof.
4. Conclusion
Firstly, we give the new concepts of sequence shadowing property and regularly recurrent point of double inverse limit space in this paper. Secondly, let (X, d) be a compact metric space, f : X⟶X be a homeomorphic map, g : X⟶X be a homeomorphic map, and f∘g = g∘f. By using the method of mutual disclosure of the double self-map f∘g and the double shift map σf∘σg, we can obtain the following results: (1) ; (2) the double self-map f∘g has sequence shadowing property if and only if the double shift map σf∘σg has sequence shadowing property. These results enrich the conclusions of sequence has shadowing property and regularly recurrent point in double inverse limit space. Most importantly, it provided the theoretical basis and scientific foundation for the application of sequence shadowing property in computational mathematics and biological mathematics.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This research was partially supported by the NSF of Guangxi Province (2020JJA110021) and construction project of Wuzhou University of China (2020B007).
