# On the Topological Dynamics of Dynamical Manifolds and Their Fundamental Group

Authors: M. Abu-Saleem; Omar Almallah
Journal: Journal of Mathematics
 (2021)

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## Abstract

The paper aims to deduce the relation between the category of topology and algebra from viewpoint of geometry and dynamical system. We introduce and define a dynamical manifold as a manifold associated with a time parameter. We obtain the induced chain of topological dynamics on the fundamental group from the chain of dynamical maps on a dynamical manifold. For many adjunctions in this context, we deduce the limit topological dynamics and conditional topological dynamics on the fundamental group. We use the category of commutative diagrams as chains of dynamical manifolds to deduce the chains on fundamental groups. Also, we describe how the manifold changes in a dynamical system from the view of the fundamental group.


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## Body

## 1. Introduction and Definitions

A dynamical system might be an arrangement that grows in time via the iterative application of a dynamical rule. The evolution rule shows the transformation of the real state under its conditions as well as possibly earlier conditions. The fact that state evolutions must be based on the system’s states suggests that the dynamics are recursive. A dynamical system, in particular, is not a simple input-output transformation; rather, the real states are determined by the system’s history. Input does not need to be given to the system continuously, and it may be sufficient if the input is only given as an initial state, and the system is then allowed to evolve according to its internal dynamical rule, and this will characterize the normal model of the dynamical system. The transition rules for the dynamical system are usually determined by a set of parameters. Exploring the qualitative nature of this dependency is an important aspect of dynamical systems theory [1, 2].A dynamical system with components, a state spaceD, a time set T⊆R, being an additive semigroup, and an operator of evolution Ψ:D×T⟶D satisfies the group property, that is, Ψy,0=y and ΨΨy,t1,t2=Ψy,t1+t2,∀y∈D and t1andt2∈D [3]. If whenever each point of N has a neighborhood that is homeomorphic to an open which contained in Rn, then we say a topological space N is locally Euclidean of dimension n. However, a second countable Hausdorff space which is locally Euclidean of dimension n is called an n-dimensional topological manifold. Indeed, the only topological manifolds were considered in this paper, for the most part, call them n-dimensional manifolds or manifolds. A submanifold of a manifold N is a subset L which itself has the structure of a manifold [4]. A surface (or 2-manifold) is a space that is homeomorphic to a nonempty subset of a finite-dimensional Euclidean space and in which every point has a neighborhood homeomorphic to R2 [5, 6]. Given a topological space X with a subspace B⊆X, we say B is a retract of X if there is a continuous map r:X⟶B for which rb=b,∀b∈B [5, 7]. Given spaces X1 and X2 with a0∈X1, b0∈X2, and X1∩X2=ϕ, then the wedge sum X1⋎X2 is the quotient of X1∪X2 under the identification a0∼b0 [8, 9]. Given a closed connected two-dimensional manifold Σ and n∈N0, we define Σ minus n open disks as the result of removing the interiors of n disjoint closed disks. The result is a compact, connected 2-dimensional manifold with n boundary components, and we write Σg,n= the surface of genus g minus n open disks where gandn∈N0 [10, 11]. The relations and connections between topological branch and algebra of the dynamical system with evidence and indications were discussed in [12–15]. Poincare introduced the fundamental group through the analysis of curves on surfaces; he classified this group by studying an analytic continuation of multivalued function ψ along with a closed path p in a manifold [11]. Given a topological space X with x1∈X. Then, the set of all homotopy classes of loops in X,x1 with an operation f1f2=f1⋅f2 is called the fundamental group and denoted by π1X,x1 [9, 16–18].


## 2. On the Limit Conditional Topological Dynamics of a Dynamical Manifold

In this section, we start by defining the topological dynamics, dynamical manifold, limit topological dynamics, and limit conditional topological dynamics on a dynamical manifold, and we obtain the induced limit topological dynamics as a functor from the category of a dynamical manifold into a category of a fundamental group.Definition 1.
LetI=a,b be the interval of time; then, it can be written as t1=a,t2,…,tm,…,b where limm⟶∞tm=b. For all t∈a,b if a manifold M associate with a time t, we call it a dynamical manifold and denote it by M,t. Also, if a map ηi associated with a time t, we call it a dynamical map and denote it by ηi,t. For simply, we denoted it by ηi.Definition 2.
Suppose thatN1,t1 and N2,t2 are two dynamical manifolds. A continuous dynamical map ηi:N1,t1⟶N2,t2 is called topological dynamics, if for every dynamical path β in a dynamical manifold N1,t1, the induced path η∘β in a dynamical manifold N2,t2 is a dynamical path with variation curvature and torsion (if needed).Definition 3.
Letηi:Ni−1,ti−1⟶Ni,ti:i=1,2,…,m be a chain of topological dynamics in a dynamical manifolds; then, we define the limit topological dynamics as limm⟶∞ηmNm−1,tm−1=limm⟶∞ηmηm−1…η1N0,t0…. Also, if the limit topological dynamics reduces the dimensions of a dynamical manifold, we call it the limit conditional topological dynamical and is denoted by limm⟶∞ξmNn,tmn=Nn−1,tn−1 where Nn,tmn is an n-dimensional dynamical manifold.Theorem 1.
Given a dynamical manifoldN0,t0. Then, there is a chain of topological dynamics in dynamical manifolds ηi:Ni−1,ti−1⟶Ni,ti:i=1,2,…,m which induces η^i:π1Ni−1,ti−1⟶π1Ni,ti:i=1,2,…m for which limm⟶∞η^mπ1Nm−1,tm−1=π1limm⟶∞ηmNm−1,tm−1.Proof.
The chain of topological dynamicsηi:Ni−1,ti−1⟶Ni,ti:i=1,2,…,m induces η^i:π1Ni−1,ti−1⟶π1Ni,ti:i=1,2,…,m for which limm⟶∞η^mπ1Nm−1,tm−1=limm⟶∞ηmα,tm−1:whereα,tm−1is  a  dynamical  loop  based  atx0,tm−1∈Nm−1,tm−1=limm⟶∞ηmα,tm−1:whereηmα,tm−1is  a  composite  loop based atηmx0,tm−1∈ηmNm−1,tm−1=limm⟶∞ηmα,tm−1:wherelimm⟶∞ηmα,tm−1is  a  dynamical  loop  based  atlimm⟶∞ηmx0,tm−1∈limm⟶∞ηmNm−1,tm−1=limm⟶∞ηmηm−1…η1N0,t0…=π1limm⟶∞ηmNm−1,tm−1.Figure 1
The chain of conditional topological dynamics on 1-dimensional manifold.Figure 2
The chain of topological dynamics on 1-dimensional manifold.Theorem 2.
LetC0,t0 be a compact connected dynamical 1-dimensional manifold without boundary. Then,(i)
The chain of conditional topological dynamicsξi:Ci−1,ti−1⟶Ci,ti:i=1,2,…,m induces a chain of conditional topological dynamics ξ^i:π1Ci−1,ti−1⟶π1Ci,ti:i=1,2,…,m for which limm⟶∞ξ^mπ1Cm−1,tm−1=0(ii)
The chain of topological dynamicsηi:Ci−1,ti−1⟶Ci,ti:i=1,2,…,m induces a chain of topological dynamics η^i:π1Ci−1,ti−1⟶π1Ci,ti:i=1,2,…,m for which limm⟶∞η^mπ1Cm−1,tim−1=0Proof.
(i)
We know thatC0,t0, the compact connected dynamical 1-dimensional manifold without boundary, is homeomorphic to S1,t0, and so, we obtain the following chain of conditional topological dynamics η1:C0,t0⟶C1,t1,η2:C1,t1⟶C2,t2,…,ηm:Cm−1,tm−1⟶Cm,tm for which limm⟶∞ηmCm−1,tm−1 is a point, as shown in Figure 1, which induces η^1:π1C0,t0⟶π1C1,t1,η^2:π1C1,t1⟶π1C2,t2,…,η^m:π1Cm−1,tm−1⟶π1Cm,tm for which limm⟶∞η^mπ1Cm−1tm−1=π1limm⟶∞ηmCm−1tm−1=π10−dimensional  dynamical  manifold. Hence, limm⟶∞ξ^π1Cm−1,tm−1=0.(ii)
Consider the chain of topological dynamicsξ1:B0,t0⟶B1,t1,ξ2:B1,t1⟶B2,t2,…,ξm:Bm−1,tm−1⟶Bm,tm, for which limm⟶∞ξmBm−1,tm−1=R1, as shown in Figure 2, which induces ξ^1:π1B0,t0⟶π1B1,t1,ξ^2:π1B1,t1⟶π1B2,t2,…,ξ^m:π1Bm−1,tm−1⟶π1Bm,tm, in which limm⟶∞ξ^π1Bm−1,tm−1=π1limm⟶∞ξmBm−1,tm−1=π1R1.
Hence,limm⟶∞ξ^π1Bm−1,tm−1=0.Theorem 3.
Given thei-surface Σ0,ni,1≤n≤2. Then, the two chains of topological dynamics ηij:Σ0,ni−1,ti−1j⟶Σ0,ni,tij:i=1,2,…,m,j=1,2 induce two chains of topological dynamics η^ij:π1Σ0,ni−1,ti−1j⟶π1Σ0,nitij:i=1,2,…,m,j=1,2 for which rank limm⟶∞η^ijπ1Σ0,nm−1,tm−1j is a free abelian group of rank two or identity group.Proof.
First, letη11:Σ0,n0,t11⟶Σ0,n1,t11,η21:Σ0,n1,t21⟶Σ0,n2,t21,…,ηm1:Σ0,nm−1,tm−11⟶Σ0,nm,tm1 be a chain of topological dynamics for which limm⟶∞ηm1Σ0,nm−1,tm−11=Σ0,0, as shown in Figures 3 and 4, which induces a chain η^11:π1Σ0,n0,t01⟶π1Σ0,n1,t11, η^21:π1Σ0,n1,t11⟶π1Σ0,n2,t21,…, η^m1:π1Σ0,nm−1,tm−11⟶π1Σ0,nm,tm1 such that limm⟶∞η^m1π1Σ0,nm−1,tm−11=π1limm⟶∞ηm1Σ0,nm−1,tm−11, and so, limm⟶∞η^mπ1Σ0,nm−1,tm−11=π1Σ0,0=0.
Also, the other chain of topological dynamicsη12:Σ0,n0,t02⟶Σ0,n1,t12,η22:Σ0,n1,t12⟶Σ0,n2,t22,…,ηm2:Σ0,nm−1,tm−12⟶Σ0,nm,tm2 for which limm⟶∞ηm2Σ0,nm−1,tm−2=T1, as shown in Figure 5, which induces a chain η^12:π1Σ0,n0,t02⟶π1Σ0,n1,t12, η^22:π1Σ0,n1,t12⟶π1Σ0,n2,t22,…, η^m2:π1Σ0,nm−1,tm−12⟶π1Σ0,nm,tm2 such that limm⟶∞η^m2π1Σ0,nm−1,tm−12=π1limm⟶∞ηm2Σ0,nm−1,tm2, and so, π1Σ0,1=π1T1. Hence, limm⟶∞η^m2π1Σ0,nm−1,tm2 is a free abelian group of rank 2. Consequently, limm⟶∞η^mjπ1Σ0,nm−1,tm2 is a free abelian group of rank two or identity group.Figure 3
Type 1: the chain of topological dynamics on a surface.Figure 4
Type 2: the chain of topological dynamics on a surface.Figure 5
Type 3: the chain of topological dynamics on a surface.Theorem 4.
(i)
Given a compact connected 1-dimensional topological dynamical manifold with boundaryJ and N0=J⋎J. Then, there is a chain of topological dynamics for which rank limm⟶∞η^mπ1Nm−1,tm−1=2(ii)
LetC0=r,θ,a≤r≤b; then, there is a chain of topological dynamics ηi:Ci−1,ti−1⟶Ci,ti:i=1,2,…,m, which induces a chain ηi,ti:π1Ci−1,ti−1⟶π1Ci,ti:i=1,2,…,m such that limm⟶∞η^mπ1Cm−1,tm−1=0Proof.
(i)
LetJ be a compact connected 1-dimensional topological manifold with boundary; then, J≈a,b, and so, N0≈a,b⋎a,b. Now, consider the following chain of topological dynamics η1:N0,t0⟶N1,t1,η2:N1,t1⟶N2,t2,…,ηm:Nm−1,tm−1⟶Nm,tm, for which limm⟶∞η1Nm−1,tm−1=S1⋎S1, as shown in Figure 6; thus, limm⟶∞η^mπ1Nm−1,tm−1=π1limm⟶∞η^mNm−1,tm−1=Z∗Z. Consequently, rank limm⟶∞η^mπ1Nm−1,tm−1=2.(ii)
LetA=R,Θ:a≤R≤b and C=R,Θ:0≤R≤b. Also, consider the following chain of topological dynamics η1:A0,t0⟶A1,t1,η2:A1,t1⟶A2,t2,…,ηm:Am−1,tm−1⟶Am,tm, for which limm⟶∞ηmAm−1,tm−1=C, as shown in Figure 7, which induces η^1: π1A0,t0⟶π1A1,t1, η^2: π1A1,t1⟶π1A2,t2,…,η^m: π1Am−1,tm−1⟶π1Am,tm for which limm⟶∞η^1π1Am−1,tm−1=π1limm⟶∞η^mAm−1,tm=π1C. Consequently, limm⟶∞η^mπ1Am−1,tm−1=0.Figure 6
The limit topological dynamical on a wedge sum of connected 1-dimensional manifold.Figure 7
The chain of topological dynamics on an annulus.Theorem 5.
IfN0=R∪Σ0,n×Z/∼ where given m∈Z, we identify m∈R with 0,0,−1,m∈Σ0,0×Z. Then, ∀k∈N, and the topological dynamics ηk:Nk−1,tk−1⟶Nk,tk induces η^k:π1Nk−1,tk−1⟶π1Nk,tk for which limk⟶∞ηkπ1Nk−1,tk−1=0.Proof.
Suppose thatN0=R∪Σ0,n×Z/∼. Also, consider the following chain of topological dynamics η1:N0,t0⟶N1,t1,η2:N1,t1⟶N2,t2,…,ηk:Nk−1,tk−1⟶Nk,tk, for which limk⟶∞ηkNk−1,tk−1=R∪Σ0,0×Z/∼, as shown in Figure 8, which induces η^1: π1N0,t0⟶π1N1,t1,η^2:N1,t1⟶N2,t2,…,η^k: π1Nk−1,tk−1⟶π1Nk,tk, for which limk⟶∞η^kπ1Nk−1,tk−1=π1limk⟶∞ηkNk−1,tk=π1R∪Σ0,0×Z/∼. Since R∪Σ0,0×Z/∼ and ⋎i=1∞Σ0,0i are of the same homotopy type, it follows that limk⟶∞ηkπ1Nk−1=0.Figure 8
The chain of topological dynamics onR∪Σ0,n×Z/∼.


## 3. The Limit Conditional Topological Dynamics on ann-Dimensional Dynamical Manifold

We now generalized the results for conditional topological dynamics on ann-dimensional dynamical manifold and describe a commutative diagram of induced homomorphism.Theorem 6.
Suppose thatNn,tn is an n-dimensional dynamical manifold and Ln,tn is a subdynamical manifold of Nn, and let ξijandrij,iandj=1,2,…,n, are sequences of conditional topological dynamics and dynamical retractions. Then, the chain of a commutative diagram on dynamical manifolds and subdynamical manifolds induces a chain of a commutative diagram on the fundamental groups.Proof.
Consider the chains of commutative diagrams, as shown in Table1.
Since the fundamental group is a functor, we get the induced chain of a commutative diagram, as shown in Table2.Table 1
The chains of commutative diagrams onNn,tn and Ln,tn.
Nn,tn⟶ξ11N1n,t1n⟶ξ21N2n,t2n⟶limm⟶∞ξm1Nn−1,tn−1↓r11↓r21↓r31limm⟶∞↓rm1Ln,sn⟶ξ11L1n,s1n⟶ξ21L2n,s2n⟶limm⟶∞ξm1Ln−1,sn−1Nn−1,tn−1⟶ξ12N1n−1,t1n−1⟶ξ22N2n−1,t2n−1⟶limm⟶∞ξm2Nn−2,tn−2↓r12↓r22↓r32limm⟶∞↓rm2Ln−1,sn−1⟶ξ12L1n−1,s1n−1⟶ξ22L2n−1,s2n−1⟶limm⟶∞ξm2Ln−2,sn−2⋮⋮⋮⋮⋮⋮⋮N1,t1⟶ξ1nN11,t11⟶ξ2nN21,t21⟶limm⟶∞ξmnN0,t0↓r1n↓r2n↓r3nlimm⟶∞↓rmnL1,s1⟶ξ1nL11,s11⟶ξ2nL21,s21⟶limm⟶∞ξmnL0,s0Table 2
The induced chains of commutative diagrams.
π1Nn,tn⟶ξ^11π1N1n,t1n⟶ξ^21π1N2n,t2n⟶limm⟶∞ξ^m1π1Nn−1,tn−1↓r^11↓r^21↓r^31limm⟶∞↓r^m1π1Ln,sn⟶ξ^11π1L1n,s1n⟶ξ^21π1L2n,s2n⟶limm⟶∞ξ^m1π1Ln−1,sn−1π1Nn−1,tn−1⟶ξ^12π1N1n−1,t1n−1⟶ξ^22π1N2n−1,t2n−1⟶limm⟶∞ξ^m2π1Nn−2,tn−2↓r^12↓r^22↓r^32limm⟶∞↓r^m2π1Ln−1,sn−1⟶ξ^12π1L1n−1,s1n−1⟶ξ^22π1L2n−1,s2n−1⟶limm⟶∞ξ^m2π1Ln−2,sn−2⋮⋮⋮⋮⋮⋮⋮π1N1,t1⟶ξ^1nπ1N11,t11⟶ξ^2nπ1N21,t21⟶limm⟶∞ξ^mnπ1N0,t0↓r^1n↓r^2n↓r^3nlimm⟶∞↓r^mnπ1L1,s1⟶ξ^1nπ1L11,s11⟶ξ^2nπ1L21,s21⟶limm⟶∞ξ^mnπ1L0,s0


## 4. Conclusion

The relation between the induced limit conditional topological dynamics and induced limit dynamical retractions on the fundamental group is deduced from the viewpoint of limit conditional topological dynamics and limit dynamical retractions on dynamical manifolds.



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