# The Research of Sequence Shadowing Property and Regularly Recurrent Point on the Double Inverse Limit Space

Authors: Zhan jiang Ji
Journal: Journal of Function Spaces
 (2022)

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## 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 mapσf∘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.


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## Body

## 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.
LetX,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
(1)RRσf∘σg=lim⟵<RRf∘g,f∘g>.Theorem 2.
LetX,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.
LetX,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]).
LetX,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
(2)Xf°g=x¯=xi,ji,j=−∞+∞:fmgnxi,j=xi−m,j−n,∀m,n≥0,∀i,j∈Z,
where we writeXf∘g=lim⟵<X,f∘g>.The metricd¯ in Xf∘g is defined by
(3)d¯x¯,y¯=maxi,j∈Zdxi,j,yi,j2i+j,where x¯=xi,ji,j=−∞+∞∈Xf∘g andy¯=yi,ji,j=−∞+∞∈Xf∘g.The double shift mapσf∘g:Xf∘g⟶Xf∘g is defined by
(4)σfmσgnxi,ji,j=−∞+∞=xi−m,j−ni,j=−∞+∞.Thus,Xf°g,d¯ is compact metric space, and the double shift map σf∘g is homeomorphic. In addition, we have
(5)σf∘σg=σg∘σf.For anyi,j∈Z, the projection map πi,j:Xf∘g⟶X is defined by
(6)πi,ixi,ji,j=−∞+∞=xi,j.Then, the mapπi,j is a continuous, surjective, and open map.Definition 5.
LetX,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 fkmx∈U, denoted by RRf the regularly recurrent point set of the map f.Remark 6.
According to the concept of regularly recurrent point of the mapf, we will give the concept of regularly recurrent point of the double map f∘g.Definition 7.
LetX,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∘glmx∈U, denoted by RRf∘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]).
LetX,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
(7)πi,j∘σfn∘σgm=fn∘gm∘πi,j.Lemma 9.
LetX,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
(8)f∘gRRf∘g⊂RRf∘g.Proof.
Supposex∈RRf∘g. Let U be an any open set containing the point f∘gx. Then, f∘g−1U is an open set containing the point x. According to x∈RRf∘g, there exists positive integer n,m>0 such that for any positive integer k,l>0, we have
(9)fkn∘glmx∈f∘g−1U.
Thus,fkn∘glmf∘gx∈U. So f∘gx∈RRf∘g. Hence,f∘gRRf∘g⊂RRf∘g.Remark 10.
The regularly recurrent point setRRf∘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 RRf∘g. Now let us give the proof process of Theorem 11.Theorem 11.
LetX,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
(10)RRσf∘σg=lim⟵<RRf∘g,f∘g>.Proof.
Supposex¯=xi,ji,j=−∞+∞∈RRσf∘σg. For any integer i,j∈Z, let U be an any open set containing the point xi,j. Thus, πi,j−1U is an open set containing the point x¯. According to x¯∈RRσf∘σg, there exists positive integer n,m>0 such that for any positive integer k,l>0, we have that
(11)σfkn∘σglmx¯∈πi,j−1U.
Thus, we have that(12)πi,j∘σfkn∘σglmx¯∈U.
By Lemma8, we can get that
(13)fkn∘glm∘πi,jx¯∈U.
That is,(14)fkn∘glmxi,j∈U.
Soxi,j∈RRf∘g. Hence, RRσf∘σg⊂lim⟵<RRf∘g,f∘g>.
Supposey¯=yi,ji,j∈−∞+∞∈lim⟵<RRf∘g,f∘g>. Then, for any integer i,j∈Z, we have
(15)yi,j∈RRf∘g.
LetV be an any open set containing the point y¯. Then, πi,jV 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
(16)fpt1∘gst2yi,j∈πi,jV.
Thus, we have that(17)πi,j−1∘fpt1∘gst2yi,j∈V.
By Lemma8, we can get that
(18)σfpt1∘σgst2∘πi,j−1yi,j∈V.
Thus, we have that(19)σfpt1∘σgst2y¯∈V.
Soy¯∈RRσf∘σg. Hence, lim⟵<RRf∘g,f∘g>⊂RRσf∘σg. This completes the proof.


## 3. Sequence Shadowing Property in the Double Inverse Limit Space

Definition 12 (see [4]).
LetX,d be a metric space and f be a continuous map from X to X. The sequence xii=0∞ is called δ-pseudo orbit of f if for any i≥0, we have dfxi,xi+1<δ.Definition 13 (see [4]).
LetX,d be a metric space and f be a continuous map from X to X. The sequence xii=0∞ is said to be ε-shadowed by the point x in X if for any i≥0, we have dfix,xi<ε.Definition 14 (see [4]).
LetX,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 xii=0∞ of f, there exists a point x in X and nonnegative integer sequence nii=0∞ such that the sequence xnii=0∞ is ε-shadowed by the pointx.Remark 15.
By Definitions12–14, we will give the concept of sequence shadowing property of the double map f∘g.Definition 16.
LetX,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 xi,ji,j=−∞+∞ is called δ-pseudo orbit of the double map f∘g if for any integer i,j∈Z, we have df∘gxi,j,xi+1,j+1<δ.Definition 17.
LetX,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 xi,ji,j=−∞+∞ is said to be ε-shadowed by the point x in X if for any positive integer i,j∈Z, we have
(20)dfi∘gjx,xi,j<ε.Definition 18.
LetX,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 xi,ji,j=−∞+∞ of f∘g, there exists a point x in X and increasing positive integer sequence nii=−∞+∞ and mij=−∞+∞ such that the sequence xni,mji,j=−∞+∞ is ε-shadowed by the point x∈X.Now we begin to prove Theorem19.Theorem 19.
LetX,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 mapf∘g has the sequence shadowing property. Since X is compact metric space, it is bounded. Write M=diamX. Then, for any ε>0, there exists positive integer m1>0 and n1>0 such that
(21)M2m1<ε,M2n1<ε.
Since the mapf and g is uniformly continuous, it follows that for any 0≤i≤2m1 and 0≤j≤2n1, there exists 0<δ1<ε such that du,v<δ1 implies
(22)dfi∘gju,fi∘gjv<ε.
Note that the double mapf∘g has the sequence shadowing property, and it follows that there exists 0<δ2<δ1 such that for any δ2-pseudo orbit xi,ji,j=−∞+∞ of f∘g, there exists a point x∈X and increasing positive integer sequence nii=−∞+∞ and mij=−∞+∞ such that the sequence xni,mji,j=−∞+∞ is δ1-shadowed by the point x. Let x¯s,ts,t=−∞+∞ be δ2/2m1+n1-pseudo orbit of the double map σf∘σg where x¯s,t=xi,js,ti,j=−∞+∞∈Xf∘g. Then, for any integer s,t∈Z, we have that
(23)d¯σf∘σgx¯s,t,x¯s+1,t+1<δ22m1+n1.
According to the definition of the metricd¯, for any integer s,t∈Z, we can get that
(24)df∘gxm1,n1s,t,xm1,n1s+1,t+12m1+n1<δ22m1+n1.
So we have that(25)df∘gxm1,n1s,t,xm1,n1s+1,t+1<δ2.
Thus,xm1,n1s,ts,t=−∞+∞ is δ2-pseudo orbit of the double map f∘g in X. Hence, there exists x∈X and increasing positive integer sequence pss=−∞+∞ and ltt=−∞+∞ such that for any integer s,t∈Z, we have that
(26)dfps∘gltx,xm1,n1ps,lt<δ1.
By (22), for any 0≤i≤2m1 and 0≤j≤2n1, we can get
(27)dfi∘gjfps∘gltx,fi∘gjxm1,n1ps,lt<ε.
Lety¯=fm1−hgn1−kxh,k=−∞+∞ and yh,k=fm1−hgn1−kx. For any i≥0 and j≥0, we have that
(28)fi∘gjyh,k=fi∘gjfm1−h∘gn1−kx=fm1−h−i∘gn1−k−jx=yh−i,k−j.
So the pointy¯ is in Xf∘g. It is easy to know that
(29)d¯σfps∘σglty¯,x¯ps,lt=maxh,k∈Zdfps∘gltfm1−h∘gn1−kx,xh,kps,lt2h+k.
Whenh>m1 or k>n1, we can, respectively, get that
(30)dfps∘gltfm1−h∘gn1−kx,xh,kps,lt2h+k<M2m1<ε,dfps∘gltfm1−h∘gn1−kx,xh,kps,lt2h+k<M2n1<ε.
Whenh>m1 and k>n1, for any 0≤i≤2m1 and 0≤j≤2n1, we have that
(31)dfps∘gltfm1−h∘gn1−kx,xh,kps,lt2h+k=dfi∘gjfps∘gltx,fi∘gjxm1,n1ps,lt2m1−i+n1−j,
wherem1−h=i and n1−k=j.
By (27), we can obtain that
(32)dfi∘gjfps∘gltx,fi∘gjxm1,n1ps,lt2m1−i+n1−j<ε.
Thus, for anyh,k∈Z we can get that
(33)dfps∘gltfm1−h∘gn1−kx,xh,kps,lt2h+k<ε.
So for any integers,t∈Z, we have that
(34)d¯σfps∘σglty¯,x¯ps,lt<ε.
Hence, the double shift mapσf∘σg has the sequence shadowing property.
Suppose the double shift map has the sequence shadowing property. For eachη>0, there exists δ3>0 such that for any δ3-pseudo orbit yi,ji,j=−∞+∞ of σf∘σg, there exists a point y∈X and increasing positive integer sequence nii=−∞+∞ and mij=−∞+∞ such that the sequence xni,mji,j=−∞+∞ is η-shadowed by the point y. Let m2>0 and n2>0 such that
(35)M2m2<δ3,M2n2<δ3.
Since the mapf and g is uniformly continuous, it follows that for any −m2≤s≤m2 and−n2≤t≤n2, there exists 0<δ4<δ3 such that du,v<δ4 implies
(36)dfs∘gtu,fs∘gtv<δ3.
Letyi,ji,j=−∞+∞ be δ4-pseudo orbit of the double map f∘g. Then, for any i,j∈Z, we have that
(37)df∘gyi,j,yi+1,j+1<δ4.
By (36), when −m2≤s≤m2 and −n2≤t≤n2, we have that
(38)dfs+1∘gt+1yi,j,fs∘gtyi+1,j+1<δ3.
For everyyi,j, let y¯i,j=f−hg−kyi,jh,k=−∞∞. It is easy to know that y¯i,j∈Xf∘g(39)d¯σf∘σgy¯i,j,y¯i+1,j+1=maxh,k∈Zdf∘gf−hg−kyi,j,f−h∘g−kyi+1,j+12h+k.
Whenh>m2 or k>n2, we can, respectively, get that
(40)df∘gf−h∘g−kyi,j,f−h∘g−kyi+1,j+12h+k<M2m2<δ3,df∘gf−h∘g−kyi,j,f−h∘g−kyi+1,j+12h+k<M2n2<δ3.
Whenh>m2 and k>n2, for any −m2≤s≤m2 and −n2≤t≤n2, we have that
(41)df∘gf−h∘g−kyi,j,f−h∘g−kyi+1,j+12h+k=dfs+1∘gt+1yi,j,fs∘gtyi+1,j+12−s+−t,
where−h=s and −k=t.
By (38), we can get that
(42)dfs+1∘gt+1yi,j,fs∘gtyi+1,j+12−s+−t<δ3.
So for any integerh,k∈Z, we have that
(43)df∘gf−h∘g−kyi,j,f−h∘g−kyi+1,j+12h+k<δ3.
Thus for any integeri,j∈Z, we have that
(44)d¯σf∘σgy¯i,j,y¯i+1,j+1<δ3.
Hence,y¯i,ji,j=−∞+∞ is δ3-pseudo orbit of the double map σf∘σg in Xf°g. So there exists a point z¯=zh,kh,k=−∞+∞∈Xf∘g and increasing positive integer sequence pii=−∞+∞ and ljj=−∞+∞ such that for any integer i,j∈Z, we have that
(45)d¯σfpi∘σgljz¯,y¯pi,lj<η.
According to the definition of the metricd¯, we can get that
(46)dfpi∘gljz00,ypi,lj<η.
Hence, the mapf∘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, letX,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) RRσf∘σg=lim⟵<RRf∘g,f∘g>; (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.



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*Source: 3512621-2022-01-12.xml*