G-Expansibility and G-Almost Periodic Point under Topological Group Action

Firstly, the new concepts of G−expansibility, G−almost periodic point, and G−limit shadowing property were introduced according to the concepts of expansibility, almost periodic point, and limit shadowing property in this paper. Secondly, we studied their dynamical relationship between the self-map f and the shift map σ in the inverse limit space under topological group action. ,e following new results are obtained. Let (X, d) be a metric G−space and (Xf, G, d, σ) be the inverse limit space of (X, G, d, f). (1) If the map f: X⟶ X is an equivalent map, then we have AP G (σ) � Lim ← (ApG(f), f). (2) If the map f: X⟶ X is an equivalent surjection, then the self-map f is G−expansive if and only if the shift map σ is G−expansive. (3) If the map f: X⟶ X is an equivalent surjection, then the self-map f has G− limit shadowing property if and only if the shift map σ has G− limit shadowing property. ,e conclusions of this paper generalize the corresponding results given in the study by Li, Niu, and Liang and Li . Most importantly, it provided the theoretical basis and scientific foundation for the application of tracking property in computational mathematics and biological mathematics.


Introduction
Let (X, d) be a metric space and f be a continuous map from X to X. A point x is called to be an almost periodic point if for each open set U containing x there exists a positive integer m > 0 such that for every positive integer k > 0 there exists r ∈ (k, k + m] satisfying f r (x) ∈ U(see [1]). In recent years, there are many achievements about almost periodic points set (see [1][2][3][4][5][6][7][8]). Qiu and Zhao [2] proved that if the map f has shadowing property in X, then the map f has shadowing property in AP(f). Liao and Wang [3] discussed the properties of almost periodic points in unilateral symbol space. In [1], it is proved that the set of almost periodic points of shift map σ is the inverse limit space formed by the self-map f in AP(f). Inspired by the idea of Li [1], the author gave the concepts of G−almost periodic point according to the concept of almost periodic point. at is, let (X, d) be a metric G−space and f be a continuous map from X to X. A point x is called to be an G−almost periodic point if for each open set U containing x there exists a positive integer m > 0 such that for every positive integer k > 0 there exists r ∈ (k, k + m] and g k ∈ G satisfying g k f r (x) ∈ U. Next, we give the following theorem in the inverse limit space under topological group action. Theorem 1. Let (X, d) be a metric G−space, the map f: X ⟶ X be an equivalent map, and (X f , G, d, σ) be the inverse limit space of (X, G, d, f). en, we have According to definition, G−almost periodic point means almost periodic point. Otherwise, it does not hold. Hence, eorem 1 generalizes the corresponding results given in Li [1]. e map f is said to be expansive if for any x ≠ y ∈ X there exists a positive integer n > 0 such that d(f n (x), f n (y)) > C (see [9]). e map f has shadowing property if each ε > 0 there exists δ > 0 such that for any δ−pseudo orbit of f there exists a point y in X such that the sequence x i ∞ i�0 is ε−shadowed by the point y (see [10]). Expansibility and shadowing property have attracted the attention of many scholars. e relevant results are seen in [11][12][13][14][15][16][17]. Das and Das [11] pointed out that the map f is G−expansive if and only if the map g is G−expansive in locally equidistant covered space. Wang and Zeng [12] discussed the relationship between average shadowing property and q −average shadowing property. Das [13] proved the nonexistence of pseudo equivariant G−expansive homeomorphism on closed unit interval. Das and Das [14] obtained a sufficient condition for the extension of a G−expansive homeomorphism on a G-invariant subspace of a compact metric G−space with G compact to be G−expansive on the whole space. Shah [15] gave a necessary and sufficient condition for a positively G−expansive map to possess the G−shadowing property. Wu [16] proved that the system with d−tracking property is chain mixed. Oprocha et al. [17] analyzed necessary and sufficient conditions for shadowing property over a set with positive density. In this paper, we gave the concepts of G−expansive map and G−limit shadowing property according to the concepts of expansive map and limit shadowing property. At last, we will give the proof of eorems 2 and 3 in the inverse limit space under topological group action. e main results are as follows.

Theorem 2.
Let (X, d) be a compact metric G−space, the map f: X ⟶ X be an equivalent surjection, and (X f , G, d, σ) be the inverse limit space of (X, G, d, f). en, the self-map f is G−expansive if and only if the shift map σ is G−expansive. Theorem 3. Let (X, d) be a compact metric G−space, the map f: X ⟶ X be an equivalent surjection, and (X f , G, d, σ) be the inverse limit space of (X, G, d, f). en, the self-map f has G−limit shadowing property if and only if the shift map σ has G−limit shadowing property.
According to definition, G−expansibility means expansibility and G−limit shadowing property means limit shadowing property. Otherwise, it does not hold. Hence, eorems 2 and 3, respectively, generalize the corresponding results given in Niu [18] and Liang and Li [10].

G-Almost Periodic Point under Topological Group Action
Definition 1 (see [19]). Let (X, d) be a metric G−space, G be a topological group, and φ: G × X ⟶ X be a continuous map. e triple (X, G, θ) or X is called to be metric G−space if the following conditions are satisfied: (1) φ(e, x) � x where e is the identity of G and for all x ∈ X (2) φ(g 1 , φ(g 2 , x)) � φ(g 1 g 2 , x) for all x ∈ X and all g 1 , g 2 ∈ G Remark 1. If X is compact, then X is also said to be compact metric G−space. For the convenience of writing, φ(g, x) is usually abbreviated as gx.
Definition 2 (see [19]). Let (X, d) be a metric G−space and f be a continuous map from X to X. f is said to be an Definition 3 (see [20]). Let (X, d) be a metric space and f be a continuous map from X to X. X f is said to be the inverse limit spaces of X if we write e metric d in X f is defined by where e projection map π i : X f ⟶ X is defined by us, (X f , d) is compact metric space. e shift mapping σ is homeomorphism, and for any i ≥ 0, the projection map π i is a continuous and open map.
Definition 4 (see [20]). Let (X, d) be a metric G-space and f be equivariant map from X to X. Write G � (g, g, g, . . .): g ∈ G , where where g � (g, g, g, . . .) ∈ G and x � ( and (X f , G, d, σ) be shown as above. e space (X f , G, d, σ) is called to be the inverse limit spaces of (X, d, f, G) under group action.    , and (X f , G, d, σ) be the inverse limit space of (X , G, d, f). en, we have is an open set containing x. Hence, there exists a positive integer m 1 > 0 such that for every positive integer k > 0 there exists p ∈ (k, k + m 1 ] and g k � (g k , g k , g k , . . .) ∈ G such that us, we can obtain en, we have that So, x i ∈ AP G (f). Hence, AP G (σ) ⊂ Lim Let t n � (t n , t n , t n , . . .) ∈ G. en, we can obtain So, y ∈ AP G (σ). us, we have Lim . is completes the proof. □

G-Expansibility under Topological Group Action
Definition 7 (see [9]). Let (X, d) be a metric space, f be a continuous map from X to X, and C be a positive constant. e map f is said to be an expansive map if for any x ≠ y ∈ X there exists a positive integer n > 0 such that d(f n (x), f n (y)) > C where the constant C is called to be an expansion constant.

Remark 3.
According to the concept of expansive map, we give the concept of G−almost expansive map. (X, d) be a metric G-space, f be a continuous map from X to X, and C be a positive constant. e map f is said to be an G−expansive map if for any x ≠ y ∈ X there exists an positive integer n such that for any g and p ∈ G we have that d(f n (gx), f n (py)) > C where the constant C is called to be an G−expansion constant.

Definition 8. Let
Next, we start to prove eorem 5. Proof. ⟹ Suppose that the self-map f is G−expansive map with expansion constant C 1 > 0. us, the map f is injective.
As the map f is injective, we can get x 0 ≠ y 0 . Let By the definition of G−expansive map f, there exists an positive integer n 1 > 0 such that By the definition of the metric d, we get that d σ n 1 (g · x), σ n 1 (l · y) > C 1 .
Hence, the shift map σ is G−expansive. ⟸ Suppose that the shift map σ is G−expansive map with expansion constant 4C 2 > 0. Since X is compact metric space, it is bounded. Write M � Diam(X). Let m > 0 such that Since the map f is surjection, for any u ≠ v ∈ X, we can choose It is obvious that the point u is different from the point v. For any s and t ∈ G, let s � (s, s, s, . . .) ∈ G, t � (t, t, t, . . .) ∈ G. (19) According to that the shift map σ is G−expansive map with expansion constant 4C 2 > 0, there exist a positive integer n 2 > 0 such that

Mathematical Problems in Engineering
According to that the map f is equivalent, we can get that m i�0 d sf n 2 +m− i (u), tf n 2 +m− i (v) By (17), we have that If for any positive integer k > 0, we have that en, by (23), we can get that m i�0 d sf n 2 +m− i (u), tf n 2 +m− i (v) It is absurd. Hence, there exists a positive integer n 3 such that d sf n 3 (u), tf n 3 (v) > C 2 . (25) So, the self-map f is G−expansive. us, we end the proof.

G-Limit Shadowing Property under Topological Group Action
Definition 9 (see [21]). Let (X, d) be a metric G−space and f be a continuous map from X to X. e sequence x i i ≥ 0 is called to be G−limit pseudo orbit of f if for any i ≥ 0 there exists t i ∈ G such that lim i⟶∞ d(t i f(x i ), x i+1 ) � 0.
Definition 10 (see [21]). Let (X, d) be a metric G−space and f be a continuous map from X to X. x i i ≥ 0 is said to be G−limit shadowed by a point y if for any i ≥ 0 there exists t i ∈ G such that lim i⟶∞ d(f i (y), t i x i ) � 0.
Definition 11 (see [21]). Let (X, d) be a metric G−space and f be a continuous map from X to X. e map f has G−limit shadowing property if for any G−limit pseudo orbit x i