# G-Chain Mixing and G-Chain Transitivity in Metric G-Space

Authors: Zhanjiang Ji
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

Firstly, we introduce the concept ofG-chain mixing, G-mixing, and G-chain transitivity in metric G-space. Secondly, we study their dynamical properties and obtain the following results. (1) If the map f has the G-shadowing property, then the map f is G-chain mixed if and only if the map f is G-mixed. (2) The map f is G-chain transitive if and only if for any positive integer k≥2, the map fk is G-chain transitive. (3) If the map f is G-pointwise chain recurrent, then the map f is G-chain transitive. (4) If there exists a nonempty open set U satisfying GU=U, U¯≠X, and fU¯⊂U, then we have that the map f is not G-chain transitive. These conclusions enrich the theory of G-chain mixing, G-mixing, and G-chain transitivity in metric G-space.


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## Body

## 1. Introduction

Chain mixing, mixing, and chain transitivity are very important concepts in topological dynamical systems. Many scholars studied their dynamical properties and obtained some valuable results (see [1–9]). Fatehi [1] showed that chain mixing, chain transitivity, and chain recurrence properties are equivalent in iterated function systems; Ji [2] proved that if the sequence map fn is G-mixed, then the limit map f is also G-mixed under G-strong uniform convergence; Zeng et al. [3] gave that if the sequence map fn is weakly mixed, then the limit map f is also weakly mixed under strong uniform convergence; Liu and Yi [4] pointed out that if the map F is chain mixed, then Fk is chain mixed for any positive integer k in nonautonomous dynamical system; Diao [5] proved that the map f is chain mixed if and only if there exists positive integer r and s such that fr×fs is chain transitive.In this paper, firstly, we introduce the concept ofG-chain mixing, G-mixing, and G-chain transitivity in metric G-space. Secondly, we study their dynamical properties and obtain the following theorem.Theorem 1.
LetX,d be a compact metric G-space, f:X⟶X be a pseudo equivalent, and isometric map and the metric d be invariant to G. If the map f has the G-shadowing property, then the map f is G-chain mixed if and only if the map f is G-mixed.Theorem 2.
LetX,d be a compact metric G-space, f:X⟶X be a pseudo equivalent map, and topological group G be compact. Then, the map f is G-chain transitive if and only if for any positive integer k≥2, the map fk is G-chain transitive.Theorem 3.
LetX,d be a compact metric G-space, f:X⟶X be a continuous map, and topological group G be compact. If the map f is a G-pointwise chain recurrent, then the map f is G-chain transitive.Theorem 4.
LetX,d be a compact metric G-space, f:X⟶X be a continuous closed map, and topological group G be compact. If there exists a nonempty open set U satisfying GU=U, U¯≠X, and fU¯⊂U, then we have that the map f is not G-chain transitive.The above conclusions enrich the theory ofG-chain mixing, G-mixing, and G-chain transitivity in metric G-space.Next, we prove Theorem1 in Section 2 and Theorems 2–4 in Section 3.


## 2.G-Chain Mixing and G-Mixing in Metric G-Space

Definition 5 (see [10]).
LetX,d be a metric G-space, G be a topological group, and φ:G×X⟶X be a continuous map. The X,G,θ or X is called a metric G-space if the following conditions are satisfied:
(1)
φe,x=x where e is the identity of G and for all x∈X(2)
φg1,φg2,x=φg1g2,x where for all x∈X and all g1, g2∈G
IfX is compact, then X is also said to be compact metric G-space. For the convenience of writing, φg,x is usually abbreviated as gx.Definition 6 (see [11]).
LetX,d be a metric G-space and f be a continuous map from X to X. f is said to be a pseudo equivariant map if for all x∈X and all p∈G, there exists p′∈G such that
(1)fpx=p′fx.Definition 7 (see [12]).
LetX,d be a metric G-space. The metric d is said to be invariant to the topological group G provided that dx,y=dgx,gy for all x,y∈X and g∈G.Definition 8 (see [7]).
LetX,d be a metric space and f be a continuous map from X to X. The map f is said to be an isometry map if we have dfx,fy=dx,y for all x,y∈X.Definition 9 (see [11]).
LetX,d be a metric G-space and f be a continuous map from X to X. The sequence xii=0∞ is called to be a G,δ-pseudo orbit of f if for any i≥0, there exists ti∈G such that dtifxi,xi+1<δ.Definition 10 (see [11]).
LetX,d be a metric G-space and f be a continuous map from X to X. The sequence xii=0∞ is said to be G,ε-shadowed by a point y in X if for any i≥0, there exists ti∈G such that dfiy,tixi<ε.Definition 11 (see [11]).
LetX,d be a metric G-space and f be a continuous map from X to X. The map f has G-shadowing property if any ε>0, there exists δ>0 such that for any G,δ-pseudo orbit xii=0∞ of f, there exists a point y in X such that the sequence xii=0∞ is G,ε-shadowed by the point y.Definition 12.
LetX,d be a metric G-space and f be a continuous map from X to X. The map f is said to be G-mixed if for any nonempty open set U and V in X, there exists a positive integer N such that for any positive integer n greater than or equal to N, there exists gn∈G such that
(2)gnfnU∩V≠∅.Definition 13 (see [13]).
LetX,d be a metric G-space and f be a continuous map from X to X. Write x0=x and y0=y. The sequence xii=0n is called to be a G,δ-chain with length n from x to y under the action of f if for any 0≤i<n, there exists gi∈G such that
(3)dgifxi,xi+1<δ.Definition 14.
LetX,d be a metric G-space and f be a continuous map from X to X. The map f is said to be G-chain mixed if for any ε>0 and x,y∈X, there exists a positive integer N such that for any positive integer n greater than or equal to N, there exists a G,δ-chain xii=0n with length n from x to y under the action of f.Now we start to prove Theorem15.Theorem 15.
LetX,d be a compact metric G-space, f:X⟶X be a pseudo equivalent and isometric map, and the metric d be invariant to G. If the map f has the G-shadowing property, then the map f is G-chain mixed if and only if the map f is G-mixed.Proof.
Suppose that the mapf is G-chain mixed. For any x,y∈X, let Bx,r1 andBx,r2 be any spherical field in X. According to that the map f has the G-shadowing property, for any 0<ε<minr1,r2, there exists 0<δ<ε such that for any G,δ-pseudo orbit xii=0∞, there exists z∈X and gi∈G such that for any i≥0, we have that
(4)dfiz,gixi<ε.
Since the mapf is G-chain mixed, there exists a positive integer N1 such that for any positive integer n greater than or equal to N1, there exists a G,δ-chain with length n from x to y under the action of f. By (4), there exists z∈X and g0,gn∈G such that
(5)dz,g0x<ε,dfnz,gny<ε.
Since the metricd is invariant to G, we can get that
(6)dg0−1z,x<ε,dgn−1fnz,y<ε.
Hence, we have that(7)gn−1fnz∈By,r2,dg0−1z,x<r1.
Combined with the definition of isometric mapf, we can obtain that
(8)dfng0−1z,fnx=dg0−1z,x.
So we can get that(9)dfng0−1z,fnx<r1.
Combined with the definition of the pseudo equivariant mapf, there exists g0−1′∈G such that
(10)dg0−1′fnz,fnx<r1.
Lettn∈G satisfying tng0−1′=gn−1. According to that the metric d is invariant to G, we have that
(11)dgn−1fnz,tnfnx<r1.
So(12)gn−1fnz∈tnfnBx,r1.
Thus,(13)tnfnBx,r1∩By,r2≠∅.
Hence, the mapf is G-mixed.
Suppose that the mapf is G-mixed. For any η>0 and x,y∈X, there exists a positive integer N2 such that for any positive integer n greater than or equal to N2, there exists tn∈G such that
(14)tnfnBx,η2∩By,η2≠∅.Let
(15)u∈tnfnBx,η2∩By,η2.
Then, we have that(16)dy,u<η2.
In addition, there existsz′∈Bx,η/2 such that u=tnfnz′. Since the metric d is invariant to G and the map f is an isometric map, we can get that
(17)dtnfnz′,tnfnx<η2.
So(18)du,tnfnx<η2.
Hence, we have that(19)dtnfnx,y<dtnfnx,u+du,y<η.
Thus,x,fx,f2x⋯fn−1x,y is a G,η-chain with length n from x to y under the action of f. So the map f is G-chain mixed. This completes the proof.


## 3.G-Chain Transitive in Metric G-Space

Definition 16 (see [13]).
LetX,d be a metric space and f be a continuous map from X to X. The CEGx,f is called to be a G-chain equivalent set of the point x if we write
(20)CEGx,f=y∈X,x⟶chGfy,y⟶chGfx.Definition 17.
LetX,d be a metric G-space and f be a continuous map from X to X. The map f is said to be G-chain transitive if for any ε>0 and x,y∈X, there exists a G,ε-chain xii=0m from x to y under the action of f.Definition 18 (see [14]).
LetX,d be a metric G-space and f be a continuous map from X to X. A point x is called to be a G-chain recurrent point if for any ε>0, there exists G,ε-chain xii=0m from x to x under the action of f, denoted by CRGf the G-chain recurrent point set of the map f.Definition 19.
LetX,d be a metric G-space and f be a continuous map from X to X. if CRGf=X, then f is said to be G-pointwise chain recurrent.Definition 20 (see [12]).
LetX,d be a metric G-space and A⊂X. Write
(21)GA≡gx:g∈G,x∈X.Now we give Lemma21 in order to prove Theorems 2 and 3 in this section.Lemma 21 (see [12]).
LetX,d be a compact metric G-space and G be compact topological group. Then, for any ε>0, there exists 0<δ<ε such that dx,y<δ implies dgx,gy<ε for any g∈G.Theorem 22.
LetX,d be a compact metric G-space, f:X⟶X be a pseudo equivalent map, and topological group G be compact. Then, the map f is G-chain transitive if and only if for any positive integer k≥2, the map fk is G-chain transitive.Proof.
Suppose that the mapf is G-chain transitive. By Lemma 21, for any ε>0, there exists a positive integer m1 greater than or equal to k such that dx,y<ε/m1 implies
(22)dgx,gy<εkforanyg∈G.
According to the uniform continuity of the mapf, for ε/m1>0, there exists a positive integer m2 greater than or equal to m1 such that dx,y<ε/m2 implies
(23)dfix,fiy<εm1forany0≤i<k.
Since the mapf is G-chain transitive, for any x,y∈X, there exists G,ε/m2-chain xii=0n from x to y under the action of f. So we have that x0=x and xn=y. Now we have two cases.Case 1.
Ifk is a factor of n, then there exists a positive integer m3 such that n=m3k. According to that xii=0n is G,ε/m2-chain under the action of f, for any 0≤i<n, there exists gi∈G such that
(24)dgifxi,xi+1<εm2.Hence, for any0≤j≤m3−1, we have that
(25)dgjkfxjk,xjk+1<εm2,dgjk+1fxjk+1,xjk+2<εm2⋯⋯dgjk+k−2fxjk+k−2,xjk+k−1<εm2,dgjk+k−1fxjk+k−1,xj+1k<εm2.Combined with the definition of pseudo equivariant mapf and (23), there exists gjk′,gjk+1′,gjk+2′⋯gjk+k−2′,gjk+k−1′∈Ggjk+k−1′=gjk+k−1 such that
(26)dgjk′fkxjk,fk−1xjk+1<εm1,dgjk+1′fk−1xjk+1,fk−2xjk+2<εm1⋯⋯dgjk+k−2′f2xjk+k−2,fxjk+k−1<εm1,dgjk+k−1′fxjk+k−1,xj+1k<εm1.By (22), we can obtain that
(27)dgjk+k−1′gjk+k−2′gjk+k−3′⋯gjk+2′gjk+1′gjk′fkxjk,gjk+k−1′gjk+k−2′gjk+k−3′⋯gjk+2′gjk+1′fk−1xjk+1<εk,dgjk+k−1′gjk+k−2′gjk+k−3′⋯gjk+2′gjk+1′fk−1xjk+1,gjk+k−1′gjk+k−2′gjk+k−3′⋯gjk+2′fk−2xjk+2<εk⋯⋯dgjk+k−1′gjk+k−2′f2xjk+k−2,gjk+k−1′fxjk+k−1<εk,dgjk+k−1′fxjk+k−1,xj+1k<εk.Hence, we can get that(28)dgjk+k−1′gjk+k−2′gjk+k−3′⋯gjk+2′gjk+1′gjk′fkxjk,xj+1k<dgjk+k−1′gjk+k−2′gjk+k−3′⋯gjk+2′gjk+1′gjk′fkxjk,gjk+k−1′gjk+k−2′gjk+k−3′⋯gjk+2′gjk+1′fk−1xjk+1+dgjk+k−1′gjk+k−2′gjk+k−3′⋯gjk+2′gjk+1′fk−1xjk+1,gjk+k−1′gjk+k−2′gjk+k−3′⋯gjk+2′fk−2xjk+2+⋯⋯+dgjk+k−1′gjk+k−2′f2xjk+k−2,gjk+k−1′fxjk+k−1+dgjk+k−1′fxjk+k−1,xj+1k<εk+εk+εk+⋯+εk=ε,so x0,xk,x2k⋯xm3k is a G,ε-chain from x to y under the action of fk. Hence, the map fk is G-chain transitive.Case 2.
Ifk is not a factor of n, then there exists a positive integer m4 such that
(29)n=km4+r,1≤r≤k−1.By the above same way, we can get thatx0,xk,x2k⋯xm4k is a G,ε-chain under the action of fk. In addition, we have that
(30)dgm4kfxm4k,xm4k+1<εm2,dgm4k+1fxm4k+1,xm4k+2<εm2⋯⋯dgm4k+r−2fxm4k+r−2,xm4k+r−1<εm2,dgm4k+r−1fxm4k+r−1,xm4k+r<εm2.According to the definition of pseudo equivariant mapf and (23), there exists tm4k,tm4k+1,tm4k+2⋯tm4k+r−2,tm4k+r−1tm4k+r−1=gm4k+r−1∈G such that
(31)dtm4kfkxm4k,fk−1xm4k+1<εm1,dtm4k+1fk−1xm4k+1,fk−2xm4k+2<ε/m1⋯⋯dtm4k+r−2f2xm4k+r−2,fxm4k+r−1<ε/m1,(32)dtm4k+r−1fxm4k+r−1,xm4k+r<εm1.By (22), we can obtain that.
(33)dtm4k+r−1tm4k+r−2tm4k+r−3⋯tm4k+2tm4k+1tm4kfkxm4k,tm4k+r−1tm4k+r−2tm4k+r−3⋯tm4k+2tm4k+1fk−1xm4k+1<εkdtm4k+r−1tm4k+r−2tm4k+r−3⋯tm4k+2tm4k+1fk−1xm4k+1,tm4k+r−1tm4k+r−2tm4k+r−3⋯tm4k+2fk−2xm4k+2<εk⋯⋯dtm4k+r−1tm4k+r−2f2xm4k+r−2,tm4k+r−1fxm4k+r−1<εkdtm4k+r−1fxm4k+r−1,xm4k+r<εk.Hence, we can get that(34)dtm4k+r−1tm4k+r−2tm4k+r−3⋯tm4k+2tm4k+1tm4kfkxm4k,xm4k+r<dtm4k+r−1tm4k+r−2tm4k+r−3⋯tm4k+2tm4k+1tm4kfkxm4k,tm4k+r−1tm4k+r−2tm4k+r−3⋯tm4k+2tm4k+1fk−1xm4k+1++dtm4k+r−1tm4k+r−2tm4k+r−3⋯tm4k+2tm4k+1fk−1xm4k+1,tm4k+r−1tm4k+r−2tm4k+r−3⋯tm4k+2fk−2xm4k+2+⋯⋯+dtmk+r−1tmk+r−2f2xm4k+r−2,tmk+r−1fxm4k+r−1+dtm4k+r−1fxm4k+r−1,xm4k+r<εk+εk+εk+⋯+εk=rεk<ε,so x0,xk,x2k⋯xm4k,xm4k+r is a G,ε-chain from x to y under the action of fk. Hence, the map fk is G-chain transitive.Suppose that the mapfk is G-chain transitive. For any η>0 and x,y∈X, there exists G,η-chain xii=0l from x to y under the action of fk. Let
(35)ymk+i=fixm,0≤m<l,0≤i≤k−1.Then, yj=0lk is a G,η-chain from x to y under the action of f. Hence, f is G-chain transitive. Thus, we end the proof.Theorem 23.
LetX,d be a compact metric G-space, f:X⟶X be a continuous map, and topological group G be compact. If the map f is a G-pointwise chain recurrent, then the map f is G-chain transitive.Proof.
Suppose that the mapf is G-pointwise chain recurrent. Let x,z∈X. According to [13], the CEGx,f is a closed set. Suppose y∈CEGx,f. By Lemma 21, for any ε>0, there exists 0<ε0<ε such that du,v<ε0/2 implies
(36)dgu,gv<ε2foranyg∈G.
According toy∈CEGx,f, there exists G,ε/2-chain under the action of f(37)x0,x1⋯xn−1,xn,y1,y2⋯ym−1,ym,where x0=ym=x and xn=y. Hence, there exists gn−1,gn∈G such that
(38)dgn−1fxn−1,y<ε2,(39)dgnfy,y1<ε2.
According to the continuity of the mapf, there exists 0<δ<ε0/2 such that dz,y<δ implies
(40)dfz,fy<ε02.
Lety′∈By,δ. By (36) and (40), we have that
(41)dgnfy,gnfy′<ε2.
According toy′∈By,δ and (38), we have that
(42)dgn−1fxn−1,y′<dgn−1fxn−1,y+dy,y′<ε.
By (39) and (41), we can get that
(43)dgnfy′,y1<dgnfy′,gnfy+dgnfy,y1<ε.
Hence,x0,x1⋯xn−1,y′,y1,y2⋯ym−1,ym is a G,ε-chain under the action of f. So we have that
(44)y′∈CEGx,f.
Thus,(45)By,δ⊂CEGx,f.
Hence, theCEGx,f is an open set. Since f is G-pointwise chain recurrent, we can get that
(46)x∈CRGf.
Hence,CEGx,f is not an empty set. Because the metric space is connected, so we have that
(47)CEGx,f=X.
Hence,(48)z∈CEGx,f.
Thus, there exists aG,ε-chain from x to y under the action of f. So the map f is G-chain transitive. Thus, we end the proof.Theorem 24.
LetX,d be a compact metric G-space, f:X⟶X be a continuous closed map, and topological group G be compact. If there exists a nonempty open set U satisfying GU=U, U¯≠X, and fU¯⊂U, then we have that the map f is not G-chain transitive.Proof.
Lety∈X−U and δ=dfU¯,X−U. Since X is compact metric space, δ is greater than 0. By Lemma 21, for the above δ/2>0, there exists 0<δ0<δ/2 such that du,v<δ0 implies
(49)dgu,gv<δ2foranyg∈G.
Letz∈fU¯. It remains to show that there is no G,δ0-chain from z to y under the action of f. If there exists a G,δ0-chain zii=0n from z to y under the action of f, then for any 0≤i≤n, there exists gi∈G such that
(50)dgifzi,zi+1<δ0.
By (49), we have that
(51)dfzi,gi−1zi+1<δ2.
In particular, we have that(52)dfz,g0−1z1<δ2.
According toz∈fU¯ and fU¯⊂U, we can obtain that
(53)fz∈fU¯.
Combined withδ=dfU¯,X−U, we can get that
(54)g0−1z1∈U.
According toGU=U, we have that the point z1 is in U. So
(55)fz1∈fU¯.By the same way, we can get zi∈U for any 2≤i≤n. Hence, y∈U. This contradicts that the point y is in X−U. So there is no G,δ0-chain from z to y under the action of f. Then, the map f is not G-chain transitive. Thus, we end the proof.


## 4. Conclusion

Firstly, we introduce the concept ofG-chain mixing, G-mixing, and G-chain transitivity in metric G-space. Secondly, we study their dynamical properties and obtain the following results. (1) If the map f has the G-shadowing property, then the map f is G-chain mixed if and only if the map f is G-mixed. (2) The map f is G-chain transitive if and only if for any positive integer k≥2, the map fk is G-chain transitive. (3) If the map f is G-pointwise chain recurrent, then the map f is G-chain transitive. (4) If there exists a nonempty open set U satisfying GU=U, U¯≠X, and fU¯⊂U, then we have that the map f is not G-chain transitive. These conclusions enrich the theory of G-chain mixing, G-mixing, and G-chain transitivity in metric G-space.Most importantly, it provided the theoretical basis and scientific foundation for the application ofG-chain mixing, G-mixing, and G-chain transitivity in computational mathematics and biological mathematics.



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