The G-Asymptotic Tracking Property and G-Asymptotic Average TrackingProperty in the Inverse Limit Spaces underGroupAction

Firstly, we introduce the definitions of G-asymptotic tracking property, G-asymptotic average tracking property, and G-quasiweak almost-periodic point. Secondly, we study their dynamical properties and characteristics. *e results obtained improve the conclusions of asymptotic tracking property, asymptotic average tracking property, and quasi-weak almost-periodic point in the inverse limit space and provide the theoretical basis and scientific foundation for the application of tracking property in computational mathematics, biological mathematics, and computer science.


Introduction
Let(X, d) be a metric space and let f: X ⟶ X be a continuous map. e sequence x i i ≥ 0 is called δ-pseudo orbit of f if, for any i ≥ 0, we have d(f(x i ), x i+1 ) < δ. e sequence x i i ≥ 0 is said to be ε-shadowed by some point y ∈ Y if, for any i ≥ 0, we have d(f(y), x i ) < ε. e map f is said to have the shadowing property if for each ε > 0 there exists δ > 0 such that, for any δ-pseudo orbit x i i ≥ 0 of f , there exists a point y ∈ X such that the sequence x i i ≥ 0 is ε-shadowed by the point y (see [1]). e shadowing property plays an important role in ergodic theory and topological dynamical systems, which has attracted the attention of many scholars in recent years. e results are shown in literature [1][2][3][4][5][6][7][8][9][10][11][12][13]. In 1980, the concept of average tracking property was introduced by Blank [1] and it was proved that some perturbed hyperbolic systems have the average tracking property. In Wang and Zeng [2], the concept of q-average tracking property is given and the q-average tracking property means chain transitivity under some conditions. Fakhari and Ghane [3] introduced the concept of ergodic tracking property and discussed its dynamical properties. Liang and Li [4] discussed the relation between the shift mapping σ and the self-mapping f about the asymptotic tracking property in the inverse limit space. By the definitions of the G-tracking property of Ekta and Tarun [10], we introduce the concepts of G-asymptotic tracking property. By the definition of the asymptotic average tracking property of Gu [13], we give G-asymptotic average tracking property. e following conclusions are obtained: (1) e self-mapping f has G-asymptotic tracking property if and only if the shift mapping σ has G-asymptotic tracking property. (2) e self-mapping f has G-asymptotic average tracking property if and only if the shift mapping σ has G-asymptotic average tracking property. us, we generalize the conclusion of Liang and Li [4]. e quasi-weak almost-periodic point is an important concept in the dynamical system, which has also attracted the attention of many scholars. e relevant results are shown by Ma [14] and Zhou and He [15]. Ma [14] proved QW(σ) � lim ← (QW(f), f). In this paper, we introduce the concept of G-quasi-weak almostperiodic point and study its topological structure in the inverse limit space under the action group. We obtain QW G (σ) � lim ← (QW G (f), f) and generalize the result of Ma [14].

G-Asymptotic Tracking Property
In this section, we will prove eorem 1. For the convenience of the reader, we give the concept used in this section. Now we start with the following definitions.
Definition 1 (see [10]). Let (X, d) be a metric space, let G be a topological group, and let θ: G × X ⟶ X be a continuous map. e triple (X, G, θ) is called a metric G-space if the following conditions are satisfied: (1) θ(e, x) � x for all x ∈ X and e is the identity of G (2) θ(g 1 , θ(g 2 , x)) � θ(g 1 g 2 , x) for all x ∈ X and all g 1 , g 2 ∈ G If (X, d) is a compact metric space, then (X, G, θ) is also said to be a compact metric G-space. For the convenience of writing, θ(g, x) is usually abbreviated as gx.
Definition 2 (see [10]). Let (X, d) be a metric G-space and let f: X ⟶ X be a continuous map. e map f is said to be an equivariant map if we have f(px) � pf(x) for all x ∈ X and p ∈ G.
Definition 3 (see [16]). Let(X, d) be a metric G-space and let f: us, (X f , d) is a compact metric space and the shift mapping σ is a homeomorphism map.
Definition 4 (see [16]). Let(X, d) be a metric G-space and let f: Let (X f , G, d, σ) and (X, G, d, f) be shown as above. e space (X f , G, d, σ) is called the inverse limit space of (X, G, d, f) under group action.
Definition 5 (see [10]). Let(X, d) be a metric G-space and let f: X ⟶ X be a continuous map. e sequence Definition 6 (see [10]). Let (X, d) be a metric G-space and let f: X ⟶ X be a continuous map. e sequence x i i ≥ 0 is said to be (G, δ)-shadowed by some point y ∈ Y if for any Definition 7 (see [10]). Let (X, d) be a metric G-space and let f: X ⟶ X be a continuous map.
e map f has G-tracking property if for each ε > 0 there exists δ > 0 such that, for any (G, δ)-pseudo orbit x i i ≥ 0 of f , there exists a point y ∈ Y such that the sequence x i i ≥ 0 is (G, ε)-shadowed by the point y.
Remarks 1. By the definitions of the G-tracking property, we will give the concept of G-asymptotic tracking property.
Now, we start to prove eorem 1.

Theorem 1.
Let (X f , G, d, σ) be the inverse limit space of (X, G, d, f) under group action. If the map f: X ⟶ X is an equivalent surjection, we have that the self-mapping f has the G-asymptotic tracking property if and only if the shift mapping σ has the G-asymptotic tracking property.
Proof. ⇒ Suppose that the map f has the G-asymptotic tracking property. Since X is compact, we write M � diam(X). For any ε > 0 , let m > 0 satisfy (M/2 m ) < (ε/2). According to the fact that the map f is uniformly continuous, for any 0 ≤ i ≤ m, there exists 0 < δ 1 < (ε/4) such that d(u, v) < δ 1 implies By the definition of G-asymptotic tracking property, for δ 1 > 0, there exists 0 < δ 2 < δ 1 and l 1 ≥ 0 such that the map f satisfies the condition of the G-asymptotic tracking property. Let y n ∞ n�0 be (G, δ 2 /2 m )-pseudo orbit, where y n � (y 0 n , y 1 n , y 2 n . . .) ∈ X f . en, for any n ≥ 0, there exists g n � (g n , g n , g n . . .) ∈ G such that d g n σ y n , y n+1 < δ 2 . (2) Hence, we have that at is, So y m n ∞ n�0 are (G, δ 2 )-pseudo orbit of the map f. us, for every n ≥ 0, there exists x ∈ X, t n ∈ G, and l 1 ≥ 0 such that d f n (x), t n y m n+l 1 < δ 1 .
According to (1) and the equivalent definition, for any n ≥ 0 and 0 ≤ i ≤ m, it follows that Because of the surjectivity of the map f, we can choose x � (f m (x), f m− 1 (x), f m− 2 (x), . . . , x, . . .) ∈ X f and t n � (t n , t n , t n . . .) ∈ G. en we have that d σ n (x), t n y n+l 1 So, the map f has the G-asymptotic tracking property. ⇐ Next we suppose that the shift mapping σ has the G-asymptotic tracking property. Let n 0 > 0. For any η > 0, there exists 0 < δ 3 < η and l 2 ≥ 0 such that, for any (G, δ 3 )-pseudo orbit x k ∞ k�1 of the shift mapping σ, we have that x k ∞ k�l2 is (G, (η/2 n 0 ))-shadowed by the point z and (M/2 n 0 ) < (δ 3 /2). Because the map f is uniformly continuous, it follows that, for any 0 ≤ i ≤ n 0 , there exists Now suppose that x k ∞ k�0 are (G, δ 4 )-pseudo orbit of the map f. en, for any k ≥ 0, there exists s k ∈ G such that According to (8) and the equivalent definition, for any k ≥ 0 and 0 ≤ i ≤ n 0 , we have that According to the surjectivity of the map f, we can choose x k � (f n 0 (x k ), f n 0 − 1 (x k ), . . . , f(x k ), x k , . . .) ∈ X f and s k � (s k , s k , s k , . . .) ∈ G. Combining G(X) � X and (10), when k ≥ 0, we have So, x k ∞ k�0 is (G, δ 3 )-pseudo orbit of the shift mapping σ. By the definition of G-asymptotic tracking property of the map σ, for any k ≥ 0, there exists z � (z 0 , z 1 , z 2 . . .) ∈ X f , l 2 ≥ 0, and p k � (p k , p k , p k . . .) ∈ G such that us, we have that Hence, we have that So the map f has the G-asymptotic tracking property. us, we end the proof.

G-Asymptotic Average Tracking Property
Definition 9 (see [17]). Let J ⊂ N. If then the set J is said to be a zero density set.
Definition 10 (see [13]). Let (X, d) be a metric space and let f: X ⟶ X be a continuous map. e sequence x i i ≥ 0 in X is called an asymptotic average pseudoorbit of the map f if Definition 11 (see [13]). Let (X, d) be a metric space and let f: X ⟶ X be a continuous map. e map f is considered to have the asymptotic average tracking property if, for any asymptotic average pseudoorbit x i i ≥ 0 , there exists a point z in X such that Remarks 2. According to the definition of asymptotic average tracking property, we will give the concept of G-asymptotic average tracking property.
Next, we give Lemma 1, which will be used in this section.
Lemma 1 (see [17]). Let a i ∞ i�0 be nonnegative real bounded sequence. en the following conclusions are equivalent: (1) lim n⟶∞ (1/n) i�n−1 i�0 a i � 0 (2) ere exists a zero density set J such that lim i∉J,i⟶∞ a i � 0 Now we will prove eorem 2 by Lemma 1.

Theorem 2.
Let (X f , G, d, σ) be the inverse limit space of (X, G, d, f) under group action. If the map f: X ⟶ X is an equivalent surjection, we have that the self-mapping f has the Discrete Dynamics in Nature and Society

G-asymptotic average tracking property if and only if the shift mapping σ has G-asymptotic average tracking property.
Proof. ⇒ Suppose that the map f has the G-asymptotic average tracking property. Since X is compact, we write M � diam(X). For any ε > 0 , let m 1 > 0 satisfy According to the fact that the map f is uniformly continuous, for any 0 ≤ k ≤ m 1 , there exists 0 Let y i ∞ i�0 be G-asymptotic average pseudo-orbit, where By Lemma 1, there exists a zero density set J 1 such that en, there exists N 1 ∈ N+ such that when i > N 1 and i ∉ J 1 , we have that us, we have that (25) at is, Hence, we have that According to Lemma 1, we have that Hence, the sequence y is G-asymptotic average pseudoorbit of the map f. By the definition of G-asymptotic average tracking property of the map f, there exist z ∈ X and t i ∈ G such that us, we can obtain r ∈ D m i . So we get that Card r: tσ r (y) ∈ B(y, η), 0 ≤ r < m i N ≥ m i .

Conclusions
In this paper, we study dynamical properties and characteristics of G-asymptotic tracking property, G-asymptotic average tracking property, and G-quasi-weak almost-periodic point.
e results obtained can generalize the conclusions of asymptotic tracking property, asymptotic average tracking property, and quasi-weak almost-periodic point in the inverse limit space. Most importantly, the paper provides the theoretical basis and scientific foundation for the application of tracking property in computational mathematics, biological mathematics, nature, and society.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.