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


               <jats:p>First, we give the concepts of <jats:italic>G</jats:italic>-sequence shadowing property, <jats:italic>G</jats:italic>-equicontinuity and <jats:italic>G</jats:italic>-regularly recurrent point. Second, we study their dynamical properties in the inverse limit space under group action. The following results are obtained. (1) The self-mapping <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0102_eq_001.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>f</m:mi>
                        </m:math>
                        <jats:tex-math>f</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> has the <jats:italic>G</jats:italic>-sequence shadowing property if and only if the shift mapping <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0102_eq_002.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>σ</m:mi>
                        </m:math>
                        <jats:tex-math>\sigma </jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> has the <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0102_eq_003.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mover accent="true">
                              <m:mrow>
                                 <m:mi>G</m:mi>
                              </m:mrow>
                              <m:mrow>
                                 <m:mo stretchy="true">¯</m:mo>
                              </m:mrow>
                           </m:mover>
                        </m:math>
                        <jats:tex-math>\overline{G}</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>-sequence shadowing property; (2) The self-mapping <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0102_eq_004.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>f</m:mi>
                        </m:math>
                        <jats:tex-math>f</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> is <jats:italic>G</jats:italic>-equicontinuous if and only if the shift mapping <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0102_eq_005.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>σ</m:mi>
                        </m:math>
                        <jats:tex-math>\sigma </jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> is <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0102_eq_006.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mover accent="true">
                              <m:mrow>
                                 <m:mi>G</m:mi>
                              </m:mrow>
                              <m:mrow>
                                 <m:mo stretchy="true">¯</m:mo>
                              </m:mrow>
                           </m:mover>
                        </m:math>
                        <jats:tex-math>\overline{G}</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>-equicontinuous; (3) <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0102_eq_007.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>R</m:mi>
                           <m:msub>
                              <m:mrow>
                                 <m:mi>R</m:mi>
                              </m:mrow>
                              <m:mrow>
                                 <m:mover accent="true">
                                    <m:mrow>
                                       <m:mi>G</m:mi>
                                    </m:mrow>
                                    <m:mrow>
                                       <m:mo stretchy="true">¯</m:mo>
                                    </m:mrow>
                                 </m:mover>
                              </m:mrow>
                           </m:msub>
                           <m:mrow>
                              <m:mo>(</m:mo>
                              <m:mrow>
                                 <m:mi>σ</m:mi>
                              </m:mrow>
                              <m:mo>)</m:mo>
                           </m:mrow>
                           <m:mo>=</m:mo>
                           <m:munder>
                              <m:mrow>
                                 <m:mi>lim</m:mi>
                              </m:mrow>
                              <m:mrow>
                                 <m:mo>←</m:mo>
                              </m:mrow>
                           </m:munder>
                           <m:mrow>
                              <m:mo>(</m:mo>
                              <m:mrow>
                                 <m:mi>R</m:mi>
                                 <m:msub>
                                    <m:mrow>
                                       <m:mi>R</m:mi>
                                    </m:mrow>
                                    <m:mrow>
                                       <m:mi>G</m:mi>
                                    </m:mrow>
                                 </m:msub>
                                 <m:mrow>
                                    <m:mrow>
                                       <m:mo>(</m:mo>
                                    </m:mrow>
                                    <m:mrow>
                                       <m:mi>f</m:mi>
                                    </m:mrow>
                                    <m:mo>)</m:mo>
                                 </m:mrow>
                                 <m:mo>,</m:mo>
                                 <m:mi>f</m:mi>
                              </m:mrow>
                              <m:mo>)</m:mo>
                           </m:mrow>
                        </m:math>
                        <jats:tex-math>R{R}_{\overline{G}}\left(\sigma )=\underleftarrow{\mathrm{lim}}\left(R{R}_{G}(f),f)</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>. These conclusions make up for the lack of theory in the inverse limit space under group action.</jats:p>


Introduction
The inverse limit space is a kind of very important space, which has always been the focus of research. However, the theory of inverse limit space has been very perfect. Scholar put forward the concept of the inverse limit space under group action and proved that the shift mapping and the self-mapping are equivariant to each other in G-shadowing property and G-strong shadowing property, see [1]. In addition, the shadowing property and equicontinuity are very important properties in the dynamical systems. Many scholars have studied their dynamical properties and obtained many meaningful results, see [2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Zhong and Wang [2] gave a sufficient and necessary condition for a point to be an equicontinuous point of dynamical system. In [3] it is shown that every ergodic invariant measure of a mean equicontinuous system has discrete spectrum; Ji, Chen and Zhang [4] proved that the shift map has the Lipschitz shadowing property if and only if the self-map has the Lipschitz shadowing property in the inverse limit space. In this paper, first, we give the concepts of G-sequence shadowing property, G-equicontinuity and G-regularly recurrent point. Second, we study their dynamical properties in the inverse limit space under group action and will get the following theorem.
Theorem A. Let X G d σ , , , f ( )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.
)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.

G-sequence shadowing property
For the convenience of the reader, we will give the concepts used in this section. Now we start with the following definitions.
) 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: where for all x X ∈ and e is the identity of G; , , for all x X ∈ and g g G , )is also said to be compact metric G-space. For the convenience of writing, θ g x , ( ) is usually abbreviated as gx.
) 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 px pf x ( ) ( ) = for all x X ∈ and p G ∈ .
) be a metric G-space and f be a continuous map from X to X.
⋯ and y y y y , , The shift mapping σ X X ) is compact metric space and the shift mapping σ is homeomorphism.
) be a metric G-space and f be an equivariant map from X to X. Write G = g g g g G , , : ) be a metric G-space and f be a continuous map from X to X. The sequence ) be a metric G-space and f be a continuous map from X to X. The sequence Remark 2.1. By Definitions 2.5 and 2.6, we will give the concept of G-sequence shadowing property.
) 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 , )-shadowed by the point y.
Now, we start to prove Theorem A.
)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, Since the map f is uniformly continuous, it follows that for any Note that the map f has the G-sequence shadowing property, it follows that there exists where y y y y X , , By (1) and the map f is equivalent, for any k 0 ≥ and Since the map f is surjective, we can choose  Since the map f is uniformly continuous, it follows that for any Suppose that 4) and the map f is equivalent, for any k 0 > and i m 0 Since the map f is surjective, for each k 0 (3) and (5) for any k 0 > , it follows that So, for any k 0 > , we have Hence, the map f has the G-sequence shadowing property. Thus, we end the proof. □ Next, we give an example satisfying G-sequence shadowing property.
Example 2.1. Let X 0, 1, , 1 n n Now, we start to prove that the map f has the G-sequence shadowing property.
Proof. It is very easy to know that X d , ( ) is a compact metric G-space and the map f is equivalent. For any )-pseudo orbit of the map f .
Hence for any i 0 ≥ there exists g G i ∈ such that d g f x x δ , .
Obviously, the distance between any two different points is greater than δ in X 1 , .
If k 0 = , according to that the map f is equivalent, we have that According to that the map f is equivalent, we have that Thus, the map f has the G-sequence shadowing property.
Case 2: For any i N ∈ , we have that Thus, we can get that Thus, the map f has the G-sequence shadowing property. □

G-equicontinuous
Let N + be the set of positive integers in this paper.
) be a metric space and f be a continuous map from X toX. 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 x f y ε , n n ( ( ) ( )) < .
Remark 3.1. According to the definition of equicontinuity, we will give the concept of G-equicontinuity.
) 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 p G , n n ∈ such that d x y δ , ( ) < implies d f g x f p y ε , n n n n ( ( ) ( )) < .

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 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 , Let δ δ 0 < and d x y δ , and y y y y X , , , .
where g g g g G , , Since the map f is uniformly continuous, it follows that for any Since the map f is surjective, we can choose that  So the map f is G-equicontinuous. This completes the proof. □ Now, we give an example satisfying G-equicontinuous. act on X by x x x x 0 ,1 1 ⋅ = ⋅ = − 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 δ ε 0 < < and g p 0   According to the definition of regularly recurrent point, we will give the concept of G-regularly recurrent point.