The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action

Theorem A

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 equivariant and surjective, we have that the self-mapping f has the G-sequence shadowing property if and only if the shift mapping σ has the G ¯ -sequence shadowing property.

Theorem B

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 equivariant and surjective, we have that the self-mapping f is G-equicontinuous if and only if the shift mapping σ is G ¯ -equicontinuous.

Theorem C

Let ( X f , G ¯ , d ¯ , σ ) be the inverse limit space of ( X , G , d , f ) under group action. If for any i ≥ 0 the map π i : X f ⟶ X is open, we have R R G ¯ ( σ ) = lim ← ( R R G ( f ) , f ) .

Definition 2.1

[16] Let ( X , d ) be a metric space, G be a topological group and θ : G × X → X be a continuous map. The triple ( X , G , θ ) is called to be a metric G-space if the following conditions are satisfied:

1. θ ( e , x ) = x , where 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 g 1 , g 2 ∈ G .

If ( X , d ) is compact, then ( X , G , θ ) is also said to be compact metric G-space. For the convenience of writing, θ ( g , x ) is usually abbreviated as g x .

Definition 2.2

[17] Let ( X , d ) be a metric G-space and f be a continuous map from X to X . The map f is said to be a equivariant map if we have f ( p x ) = p f ( x ) for all x ∈ X and p ∈ G .

Definition 2.3

[1] Let ( X , d ) be a metric G-space and f be a continuous map from X to X . lim ← ( X , f ) is said to be the inverse limit space if we write lim ← ( X , f ) = { ( x 0 , x 1 , x 2 ⋯ ) : f ( x i + 1 ) = x i , i ≥ 0 } , where lim ← ( X , f ) is denoted by X f .

The metric d ¯ in X f is defined by d ¯ ( x ¯ , y ¯ ) = ∑ i = 0 ∞ d ( x i , y i ) 2 i , where x ¯ = ( x 0 , x 1 , x 2 ⋯ ) and y ¯ = ( y 0 , y 1 , y 2 ⋯ ) . The shift mapping σ : X f → X f is defined by σ ( x ¯ ) = ( f ( x 0 ) , x 0 , x 1 ⋯ ) . Thus, ( X f , d ¯ ) is compact metric space and the shift mapping σ is homeomorphism.

Definition 2.4

[1] Let ( X , d ) be a metric G-space and f be an equivariant map from X to X . Write G ¯ = { ( g , g , g ⋯ ) : g ∈ G } and G ∞ = ∏ i = 0 ∞ G i where G i = G . The map θ : G ¯ × X f → X f is defined by θ ( g ¯ , x ¯ ) = g ¯ x ¯ = ( g x 0 , g x 1 , g x 2 ⋯ ) , where g ¯ = ( g , g , g ⋯ ) ∈ G ¯ and x ¯ = ( x 0 , x 1 , x 2 ⋯ ) ∈ X f . Then ( X f , G ¯ , θ ) is a metric G ¯ -space.

Let ( X f , G ¯ , d ¯ , σ ) and ( X , G , d , f ) be shown as above. The space ( X f , G ¯ , d ¯ , σ ) is called to be the inverse limit spaces of ( X , G , d , f ) under group action.

Definition 2.5

[18] Let ( X , d ) be a metric G-space and f be a continuous map from X to X . The sequence { x i } i ≥ 0 is called to be ( G , δ ) -pseudo orbit of f if for any i ≥ 0 there exists t i ∈ G such that d ( t i f ( x i ) , x i + 1 ) < δ .

Definition 2.6

[18] Let ( X , d ) be a metric G-space and f be a continuous map from X to X . The sequence { x i } i ≥ 0 is said to be ( G , δ ) -shadowed by a point y ∈ X if for any i ≥ 0 there exists t i ∈ G such that d ( f i ( y ) , t i x i ) < δ .

Remark 2.1

By Definitions 2.5 and 2.6, we will give the concept of G-sequence shadowing property.

Definition 2.7

Let ( X , d ) be a metric G-space and f be a continuous map from X to X . The map f has G-sequence shadowing property if 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 ∈ X and nonnegative integer sequence { n i } i ≥ 0 such that the sequence { x n i } i ≥ 0 is ( G , ε ) -shadowed by the point y .

Theorem A

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 equivalent and surjective, we have that the self-mapping f has the G-sequence shadowing property if and only if the shift mapping σ has the G ¯ -sequence shadowing property.

Proof

⇒ Suppose that the map f has the G-sequence shadowing property. Since X is compact metric space, it is bounded. Write M = diam ( X ) . Then for any ε > 0 , there exists m > 0 such that

Since the map f is uniformly continuous, it follows that for any 0 ≤ i ≤ m , there exists 0 < δ 1 < ε 4 such that d ( x , y ) < δ 1 implies

Note that the map f has the G -sequence shadowing property, it follows that there exists 0 < δ 2 < δ 1 such that any ( G , δ 2 ) -pseudo orbit { x i } i ≥ 0 of f , there exists a point y ∈ X and nonnegative integer sequence { n i } i ≥ 0 such that the sequence { x n i } i ≥ 0 is ( G , δ 1 ) -shadowed by the point y . Let { y ¯ k } k ≥ 0 be G ¯ , δ 2 2 m -pseudo orbit, where y ¯ k = ( y k 0 , y k 1 , y k 2 ⋯ ) ∈ X f . Hence for any k ≥ 0 there exists g ¯ k = ( g k , g k , g k ⋯ ) ∈ G ¯ such that

That is, for any k ≥ 0 , we have

Thus, { y k m } k ≥ 0 is ( G , δ 2 ) -pseudo orbit in X . Hence, there exists x 0 ∈ X , t k ∈ G and nonnegative integer sequence { n k } k ≥ 0 such that

By (1) and the map f is equivalent, for any k ≥ 0 and 0 ≤ i ≤ m , we have

Since the map f is surjective, we can choose s ¯ = ( f m ( x 0 ) , f m − 1 ( x 0 ) , f m − 2 ( x 0 ) ⋯ x 0 ⋯ ) ∈ X f and t ¯ k = ( t k , t k , t k ⋯ ) ∈ G ¯ . By (2), for any k ≥ 0 , it follows that

Hence, the shift mapping σ has the G ¯ -sequence shadowing property.

⇐ Suppose the shift mapping σ has the G ¯ -sequence shadowing property. Let m 0 > 0 . For each η > 0 there exists δ 3 > 0 such that for any ( G ¯ , δ 3 ) -pseudo orbit { z ¯ k } k ≥ 0 of σ , there exists a point z ∈ X and nonnegative integer sequence { n k } k ≥ 0 such that the sequence { z ¯ n k } k ≥ 0 is G ¯ , η 2 m 0 -shadowed by the point z and

Since the map f is uniformly continuous, it follows that for any 0 ≤ i ≤ m 0 there exists 0 < δ 4 < δ 3 4 such that d ( x , y ) < δ 4 implies

Suppose that { x k } k ≥ 0 is ( G , δ 4 ) -pseudo orbit in X . Thus, for any k > 0 there exists l k ∈ G such that

By (4) and the map f is equivalent, for any k > 0 and 0 ≤ i ≤ m 0 , we have

Since the map f is surjective, for each k > 0 we can choose z ¯ k = ( f m 0 ( x k ) , f m 0 − 1 ( x k ) , ⋯ x k ⋯ ) ∈ X f and l ¯ k = ( l k , l k , l k ⋯ ) ∈ G ¯ . According to (3) and (5) for any k > 0 , it follows that

Hence, { z ¯ k } k ≥ 0 is ( G ¯ , δ 3 ) -pseudo orbit in X f . Thus, there exists z ¯ = ( z 0 , z 1 , z 2 ⋯ ) ∈ X f , p ¯ k = ( p k , p k , p k ⋯ ) ∈ G ¯ and nonnegative integer sequence { n k } k ≥ 0 such that

So, for any k > 0 , we have

Hence, the map f has the G-sequence shadowing property. Thus, we end the proof.□

Example 2.1

Let X = 0 , − 1 , − 1 n , − 1 + 1 n . The metric d in X is defined by d ( x , y ) = ∣ x − y ∣ where x , y ∈ X . Let G = Z 2 = { 0 , 1 } . Defined by 0 ⋅ x = x , 1 ⋅ x = − 1 − x for every x ∈ X . The map f : X → X is defined by

Proof

It is very easy to know that ( X , d ) is a compact metric G -space and the map f is equivalent. For any η > 0 , there exists m > 0 such that 1 m < η . Write δ = 1 4 m ( m + 1 ) . Let { x i } i ≥ 0 be ( G , δ ) -pseudo orbit of the map f . Hence for any i ≥ 0 there exists g i ∈ G such that

Obviously, the distance between any two different points is greater than δ in − 1 + 1 m + 1 , − 1 m + 1 ⋂ X . Hence, we have two cases.

Case 1: There exists k ∈ N such that

According to the inequality d ( g k f ( x k ) , x k + 1 ) < δ , we have that

and

Keep going, we can get x i + 1 = g i f ( x i ) and x i ∈ − 1 + 1 m , − 1 m ⋂ X when i ≥ k . If k = 0 , according to that the map f is equivalent, we have that

If k ≥ 1 , according to the inequality d ( g k − 1 f ( x k − 1 ) , x k ) < δ , we have that

and

Keep going, we can get x i = g i − 1 f ( x i − 1 ) and x i ∈ 1 m , 1 − 1 m ⋂ X when i ≤ k . Hence, when i ≥ 1 , we have that

According to that the map f is equivalent, we have that

Hence, we have d ( g i g i − 1 ⋯ g 0 f i ( x 0 ) , x i ) = 0 < η . Thus, the map f has the G-sequence shadowing property.

Case 2: For any i ∈ N , we have that

When x i ∈ − 1 , − 1 + 1 m ⋂ X , we write g i = 1 . When x i ∈ − 1 m , 0 ⋂ X , we write g i = 0 . Thus, we can get that

Thus, the map f has the G-sequence shadowing property.□

Definition 3.1

Let ( X , d ) be a metric space and f be a continuous map from X to X . The map f is said to be equicontinuous if for any ε > 0 and n ∈ N + there exists δ > 0 such that d ( x , y ) < δ implies d ( f n ( x ) , f n ( y ) ) < ε .

Remark 3.1

According to the definition of equicontinuity, we will give the concept of G-equicontinuity.

Definition 3.2

Let ( X , d ) be a metric G-space and f be a continuous map from X to X . The map f is said to be G-equicontinuous if each ε > 0 there exists δ > 0 such that for any n ∈ N + there exists g n , p n ∈ G such that d ( x , y ) < δ implies d ( f n ( g n x ) , f n ( p n y ) ) < ε .

Remark 3.2

Let Z + be the set of nonnegative positive integers. If G = Z + , then ( X , Z + , φ ) is a semi discrete dynamical system. According to [19], there exists a continuous map f from X to X such that for any x ∈ X and m ∈ Z + , we have φ ( m , x ) = f m ( x ) . In this case, the map f has G-equicontinuous means that each ε > 0 there exists δ > 0 such that for any n ∈ N + there exists m , k ∈ Z + such that d ( x , y ) < δ implies d ( f n + m ( x ) , f n + k ( y ) ) < ε . Hence, Definition 3.2 is broader than Definition 3.1 even for G = Z + .

Theorem B

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 equivariant and surjective, we have that the self-mapping f is G-equicontinuous if and only if the shift mapping σ is G ¯ -equicontinuous.

Proof

⇒ Suppose the map f is G-equicontinuous. Hence, for any ε > 0 there exists 0 < δ < ε 4 such that for any n ≥ 0 there exists g n , p n ∈ G such that d ( x , y ) < δ implies

Let δ 0 < δ and d ¯ ( x ¯ , y ¯ ) < δ 0 , where x ¯ = ( x 0 , x 1 , x 2 ⋯ ) ∈ X f and y ¯ = ( y 0 , y 1 , y 2 ⋯ ) ∈ X f . Thus, we have

By (6), for any n ≥ 0 there exists g n , p n ∈ G such that

Let g ¯ n = ( g n , g n − 1 ⋯ g 1 , g 0 , e , e , e ⋯ ) ∈ G ¯ and k ¯ n = ( k n , k n − 1 ⋯ k 1 , k 0 , e , e , e ⋯ ) ∈ G ¯ . According to the map σ is an equivalent map, it follows that

So the shift mapping σ is G-equicontinuous.

⇐ Suppose the shift mapping σ is G ¯ -equicontinuous. For any ε > 0 , there exists 0 < δ 1 < ε 4 such that for any n ≥ 0 , there exists g ¯ n , k ¯ n ∈ G ¯ such that d ¯ ( x ¯ , y ¯ ) < δ 1 implies

where g ¯ n = ( g n 0 , g n 1 , g n 2 ⋯ ) ∈ G ¯ and k ¯ n = ( k n 0 , k n 1 , k n 2 ⋯ ) ∈ G ¯ . Since X is compact metric space, it is bounded. Write M = diam ( X ) . Let m > 0 such that

Since the map f is uniformly continuous, it follows that for any 0 ≤ i ≤ m there exists 0 < δ 2 < δ 1 4 such that d ( x , y ) < δ 2 implies

Let x 0 , y 0 ∈ X such that d ( x 0 , y 0 ) < δ 2 . By (8), we have

Since the map f is surjective, we can choose that

Hence, we have that

By (7), for any n ≥ 0 , we have that

where g ¯ n = ( g n 0 , g n 1 , g n 2 ⋯ ) ∈ G ¯ and k ¯ n = ( k n 0 , k n 1 , k n 2 ⋯ ) ∈ G ¯ . Thus, it follows that

By the map f is equivalent, we get

So the map f is G-equicontinuous. This completes the proof.□

Example 3.1

Let X = [ 0 , 1 ] . The metric d in X is defined by d ( x , y ) = ∣ x − y ∣ where x , y ∈ X . The map f : X → X is defined by f ( x ) = x . Let G = { 0 , 1 } act on X by 0 ⋅ x = x , 1 ⋅ x = 1 − x for every x ∈ X . It is very easy to know that ( X , d ) is a metric G -space. For any ε > 0 and n ∈ N + , let 0 < δ < ε and g n = p n = 0 . If d ( x , y ) < δ , then we have that

So the map f is G-equicontinuous.

Definition 4.1

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 regularly recurrent point if for each open set U containing the point x , there exists m > 0 such that for any k > 0 , we have f k m ( x ) ∈ U . Denoted by R R ( f ) the regularly recurrent point set of the map f .

Remark 4.1

According to the definition of regularly recurrent point, we will give the concept of G-regularly recurrent point.

Definition 4.2

Let ( X , d ) be a metric G-space and f be a continuous map from X to X . A point x ∈ X is called to be G-regularly recurrent point if for each open set U containing the point x , there exists m > 0 such that for any k > 0 there exists g k ∈ G such that g k f k m ( x ) ∈ U . Denoted by R R G ( f ) the G-regularly recurrent point set of the map f .

Theorem C

Let ( X f , G ¯ , d ¯ , σ ) be the inverse limit space of ( X , G , d , f ) under group action. If for any i ≥ 0 the map π i : X f ⟶ X is open, we have R R G ¯ ( σ ) = lim ← ( R R G ( f ) , f ) .