() @ Appl. Gen. Topol. 14, no. 2 (2013), 179-193doi:10.4995/agt.2013.1671 c© AGT, UPV, 2013 Discrete dynamics on noncommutative CW complexes Vida Milani a and Seyed M. H. Mansourbeigi b a Dept. of Math., Faculty of Math. Sci., Shahid Beheshti University, Tehran 1983963113, Iran. School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332, USA (v-milani@sbu.ac.ir; vmilani3@math.gatech.edu) b Dept. of Electrical Engineering, Polytechnic University, NY 11747, USA Department of Electrical Engineering, SCCC, Brentwood, NY 11717, USA (mansous@sunysuffolk.edu) Abstract The concept of discrete multivalued dynamical systems for noncommu- tative CW complexes is developed. Stable and unstable manifolds are introduced and their role in geometric and topological configurations of noncommutative CW complexes is studied. Our technique is illustrated by an example on the noncommutative CW complex decomposition of the algebra of continuous functions on two dimensional torus. 2010 MSC: 46L85, 55U10, 54H20, 34D35. Keywords: closed hemi-continuous, C*-algebra, CW complexes, discrete dy- namical system, modified Morse function, noncommutative CW complex, open hemi-continuous, stable manifold, unstable man- ifold. 1. Introduction The theory of CW complexes was invented by Whitehead in 1949 [14]. The concept of CW complex structures on topological manifolds has been a great development in the category of topological spaces [8]. It is a well known fact that the topology of a manifold can be reconstructed from the commutative C*-algebra of continuous functions on it [7, 10]. In other words commutative C*-algebras play as the dual concept for topological manifolds. Away from Received January 2013 – Accepted August 2013 http://dx.doi.org/10.4995/agt.2013.1671 V. Milani and S. M. H. Mansourbeigi commutativity, C*-algebras are still substitutes for noncommutative topologi- cal manifolds and provide building blocks of noncommutative topology theory [2, 4, 10]. In the category of noncommutative manifolds, noncommutative CW complexes were introduced in [6, 13]. A great development in the theory of noncommutative topology would be the study of noncommutative manifolds (C*-algebras) which are endowed with a noncommutative CW complex struc- ture. In the study of CW complexes there exist two classical approaches. One approach comes from differential topology and Morse theory [11]. The second one is the dynamics point of view and the relation between dynamical properties of a flow and the homological configuration of the CW complex. Our aim is to develop the two approaches in the framework of noncommutative topology in order to study noncommutative CW complexes: Dynamics Primitive Spectrum −−−−−−−−−−−−→ NCCW Complexes C*-Algebra ←−−−−−−− Diff. Topology In this regard our first attempt was the development of the Morse theory ap- proach in [12]. In the present paper we are developing the second approach. In both approaches we apply techniques from combinatorial topology [3, 5] and the primitive spectrum of C*-algebras [10] as basic tools. The paper is organized as follows. In section 2 we review fundamental no- tions in the theory of noncommutative CW complexes. We explain the role of the primitive spectrum as a bridge between CW complexes and noncommuta- tive CW complexes. Section 3 is devoted to a review from [12] on the basics of modified Morse theory on noncommutative CW complexes. Discrete multival- ued dynamical systems have been introduced in [1, 9]. In section 4 we develop discrete multivalued dynamical systems on noncommutative CW complexes and provide tools to relate a dynamical picture to the topology and geometry of noncommutative CW complexes. We will see how the dynamical proper- ties of the trajectories are related to the configuration of noncommutative CW complexes. In this section, stable and unstable manifolds are introduced and some of their properties are studied. An example will serve to illustrate our dynamical construction. Section 5 is devoted to the explanation of this exam- ple. In this section we study the noncommutative CW complex structure of C(T 2): the algebra of continuous functions on the 2-dimensional torus. We as- sociate a discrete multivalued dynamical system with it. We shall see how the configuration of this noncommutative CW complex is explained by the stable and unstable manifolds. 2. Noncommutative CW complexes In this section we review basic definitions and results on the theory of non- commutative CW complexes from [6, 13]. we explain the technique of the primitive spectrum and its role as a link between CW complexes and noncom- mutative CW complexes. Details on the structure of primitive spectrum can be found in [7, 10, 12]. First we review the concept of CW complex structure for a topological space from [8]. c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 180 Discrete dynamics on noncommutative CW complexes A sequence X0 ⊂ X1 ⊂ ... ⊂ Xn = X is an n-dimensional CW complex structure for a compact topological space X, where X0 is a finite discrete space consisting of 0-cells, and for k = 1, ...,n each k-skeleton Xk is obtained by attaching λk number of k-disks to Xk−1 via the attaching maps ϕk : ⋃ λk Sk−1 → Xk−1. In other words (2.1) Xk = Xk−1 ⋃ (∪λkI k) x ∼ ϕk(x) := Xk−1 ⋃ ϕk (∪λkI k) where Ik := [0,1]k and Sk−1 := ∂Ik. The quotient map is denoted by ρ : Xk−1 ⋃ (∪λkI k) → Xk. For a continuous map φ : X → Y between compact topological spaces X and Y , the C*-morphism induced on their associated C*-algebra of functions is denoted by C(φ) : C(Y ) → C(X) which is defined by C(φ)(g) := g ◦φ for g ∈ C(Y ). Definition 2.1. Let A1, A2 and C be C*-algebras. A pull back for C via morphisms α1 : A1 → C and α2 : A2 → C is the C*-subalgebra of A1 ⊕ A2 denoted by PB(C,α1,α2) defined by PB(C,α1,α2) := {a1 ⊕a2 ∈ A1 ⊕A2 : α1(a1) = α2(a2)}. For any C*-algebra A, let SnA := C(Sn → A),InA := C([0,1]n → A),In0 A := C0((0,1) n → A), where Sn is the n-dimensional unit sphere. Definition 2.2. A 0-dimensional noncommutative CW complex is any finite dimensional C*-algebra A0. Recursively an n-dimensional noncommutative CW complex is any C*-algebra appearing in the following diagram 0 −−−−→ In0 Fn −−−−→ An π −−−−→ An−1 −−−−→ 0 ∥ ∥ ∥   y fn   y ϕn 0 −−−−→ In0 Fn −−−−→ I nFn δ −−−−→ Sn−1Fn −−−−→ 0 Where the rows are extensions, An−1 an (n−1)-dimensional noncommutative CW complex, Fn some finite (linear) dimensional C*-algebra of dimension λn, δ the boundary restriction map, ϕn an arbitrary morphism (called the connecting morphism), for which An = PB(S n−1Fn,δ,ϕn) := {(α,β) ∈ I nFn ⊕An−1 : δ(α) = ϕn(β)}, and fn and π are respectively projections onto the first and second coordinates. c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 181 V. Milani and S. M. H. Mansourbeigi With these notations {A0, ...,An} is called the noncommutative CW complex decomposition of dimension n for A = An. For each k = 0,1, ...,n, Ak is called the k-th decomposition cell. Let A be a unital C*-algebra. The primitive spectrum of A is the space of kernels of irreducible *-representations of A. It is denoted by Prim(A). The topology on this space is given by the closure operation as follows: For any subset U ⊆ Prim(A), the closure of U is defined by U := {I ∈ Prim(A) : ⋂ J∈U J ⊂ I} Obviously U ⊆ U. This operation defines a topology on Prim(A) (the hull- kernel topology), making it into a T0-space [10]. Definition 2.3. A subset U ⊆ Prim(A) is called absorbing if it satisfies the following condition: I ∈ U,I ⊆ J ⇒ J ∈ U. Remark 2.4. The closed subsets of Prim(A) are exactly its absorbing subsets. In the special case, when M is a compact topological space, and A = C(M) is the commutative unital C*-algebra of complex continuous functions on M, a homeomorphism between M and prim(A) is obtained in the following way. For each x ∈ M let Ix := {f ∈ A : f(x) = 0}; Ix is a closed maximal ideal of A. It is in fact the kernel of the evaluation map (ev)x :A −→ C f 7−→ f(x). Now I : M → Prim(A) defined by I(x) := Ix is the desired homeomorphism. let X0 ⊂ X1 ⊂ ... ⊂ Xn = X be an n-dimensional CW complex structure for the compact space X. A cell complex structure is induced on Prim(C(X)) by the following procedure: Let Ak = C(Xk), k = 0,1, ...,n. Set A = C(X) = C(Xn) = An. Consider the homeomorphism I : X → Prim(C(X)). For each k-cell Ck in the k-skeleton Xk, let ICk = ⋂ x∈Ck Ix = {f ∈ A : f(x) = 0;x ∈ Ck}, for 0 ≤ k ≤ n . By considering the restriction of functions on X to Xk, ICk will be an ideal in Ak. In the above notations, the closed sets Wi0,...,ik := {J ∈ Prim(Ak) : J ⊇ ICk} are corresponded to the ideals ICk. c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 182 Discrete dynamics on noncommutative CW complexes In general we can have Proposition 2.5 ([12]). Let X be an n-dimensional CW complex containing cells of each dimension k = 0, ...,n. Then there exists a noncommutative CW complex decomposition of dimension n for A = C(X). Conversely if {A0, ...,An} be a noncommutative CW complex decomposition for the C*-algebra A such that Ais (i = 0, ..,n) are unital, Then there exists an n-dimensional CW complex structure on Prim(A). Example 2.6. Let X0 = {0,1}and X1 = [0,1] be the zero and the one skeleton for a CW complex structure of [0,1]. Then we have A0 = C(X0) ≃ C⊕ C and A = A1 = C(X1). The 0-ideals I0 and I1 and their corresponding 0-chains W0 and W1 are as follow: I0 = {f ∈ A0 : f(0) = 0}≃ C,I1 = {f ∈ A0 : f(1) = 0}≃ C, W0 = {J ∈ Prim(A0) : J ⊇ I0} = {I0},W1 = {J ∈ Prim(A0) : J ⊇ I1} = {I1}. Corresponding to the 1-chain C1 = [0,1], the only 1-ideal is I = ⋂ x∈C1 Ix = {f ∈ A : f(x) = 0;x ∈ [0,1]} = {0}, with the corresponding 1-chain WI = {J ∈ Prim(A) : J ⊇ I} = Prim(A) ≃ [0,1]. Proposition (2.5) can be extended to an arbitrary unital C*-algebra. Let A be an arbitrary unital C*-algebra. To each I ∈ Prim(A), there corresponds an absorbing set WI := {J ∈ Prim(A) : J ⊇ I}, and an open set OI := {J ∈ Prim(A) : J ⊆ I}, containing I. We have the following equivalent statements: I ⊆ J ⇔ OI ⊆ OJ ⇔ WI ⊇ WJ In [12] we have seen how Prim(A) is made into a finite lattice with vertices I0, ...,In. Let Ji0,...,ik := Ii0 ∩ ...∩Iik, where 1 ≤ i0, ..., ik ≤ n,1 ≤ k ≤ n.Set Wi0,...,ik := {J ∈ Prim(A) : J ⊇ Ji0,...,ik}. As we have seen in [12], these are the k-chain closed subsets of Prim(A) having the following property If Ji0,...,ik = 0 for some 1 ≤ i0, ..., ik ≤ n, 1 ≤ k ≤ n, then Wi0,...,ik = Prim(A). Also for each pair of indices (i0, ..., it) , σ(i0, ..., it+m), Wi0,...,it ⊆ Wσ(i0,...,it+m) c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 183 V. Milani and S. M. H. Mansourbeigi where σ is a permutation on t + m + 1 elements and 1 ≤ i0, ..., it+m ≤ n. In the case of Prim(C(X)) when X has a CW complex structure, the k- chains are the closed sets Wi0,...,ik = {J ∈ Prim(Ak) : J ⊇ ICk} corresponding to the k-ideals ICk [12]. 3. Basics of Modified Morse Theory on C*-Algebras The first step towards understanding the geometry of noncommutative CW complexes was the idea of modified Morse theory on C*-algebras that we have done in [12]. In this section we review some of the results. For a unital C*-algebra A let Σ = {Wi1,...,ik}1≤i1,...,ik≤n,1≤k≤n be the set of all k-chains (k = 1, ...,n) in Prim(A), and Γ = {Ii1,...,ik}1≤i1,...,ik≤n,1≤k≤n be the absorbing set of all k-ideals corresponding to the k-chains of Σ for k = 1, ...,n. We recall the following definitions from [12]. Definition 3.1. Let f : Σ → R be a function. The k-chain Wk = Wi1,...,ik is called a critical chain of order k for f, if for each (k+1)-chain Wk+1 containing Wk and for each (k −1)-chain Wk−1 contained in Wk, we have f(Wk−1) ≤ f(Wk) ≤ f(Wk+1). The corresponding ideal Ik to Wk is called the critical ideal of order k. Definition 3.2. Let f has a critical chain of order k. We say f is an acceptable Morse function, if it has a critical chain of order i, for all i ≤ k. Definition 3.3. A function f : Σ → R is called a modified Morse function on the C*-algebra A, if for each k-chain Wk in Σ, there is at most one (k+1)-chain Wk+1 containing Wk and at most one (k-1)-chain Wk−1 contained in Wk, such that f(Wk+1) ≤ f(Wk) ≤ f(Wk−1). Definition 3.4. If A, B are two C*-algebras, two morphisms α,β : A → B are homotopic, written α ∼ β,if there exists a family {Ht}t∈[0,1] of morphisms Ht : A → B such that for each a ∈ A the map t 7→ Ht(a) is a norm continuous path in B with H0 = α and H1 = β.The C*-algebras A and B are said to have the same homotopy type, if there exists morphisms ϕ : A → B and ψ : B → A such that ϕ◦ψ ∼ idB and ψ ◦ϕ ∼ idA. In this case the morphisms ϕ and ψ are called homotopy equivalence. Definition 3.5. Let A and B be unital C*-algebras. We say A is of pseudo- homotopy type as B if C(Prim(A)) and B have the same homotopy type. c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 184 Discrete dynamics on noncommutative CW complexes Remark 3.6. In the case of unital commutative C*-algebras, by the GNS con- struction [7], C(Prim(A)) = A, . So the notions of pseudo-homotopy type and the same homotopy type are equivalent. Theorem 3.7. If f is an acceptable modified Morse function on A, then Prim(A) is homotopy equivalent to a CW complex with exactly one cell of dimension p for each critical chain of order p. Consequently every unital C*- algebra A with an acceptable modified Morse function f on it, is of pseudo- homotopy type as a noncommutative CW complex having a k-th decomposition cell for each critical chain of order k. 4. Dynamical systems on noncommutative CW complexes In this section we develop tools to relate a dynamical picture to the topology and geometry of noncommutative CW complexes. We will see how the dynam- ical properties of the trajectories are related to the homological configuration of noncommutative CW complexes. Definition 4.1. Let X,Y be topological spaces, P(Y ) be the power set of Y (the set of all subsets of Y ) and F : X →P(Y ) be a mapping. • The mapping F is called open hemi-continuous at x ∈ X, if for each open subset B ⊆ Y such that F(x) ⊆ B, there exists an open set U ⊆ X containing x such that F(U) := ⋃ {F(x) : x ∈ U}⊆ B. • The mapping F is called closed hemi-continuous at x ∈ X, if for each closed subset B ⊆ Y such that F(x) ⊆ B, there exists a closed set K ⊆ X containing x such that F(K) := ⋃ {F(x) : x ∈ K}⊆ B. • The mapping F is with compact value if for all x ∈ X, F(x) ⊆ Y is a compact subset. Definition 4.2. Let X be a topological space and P(X) be the power set of X. A mapping ϕ : X ×Z →P(X) is a discrete multivalued dynamical system on X if the following conditions satisfy: • For each n ∈ Z the mapping Fn : X → P(X) defined by Fn(x) := ϕ(x,n), for all x ∈ X, is closed hemi-continuous for n ∈ Z+ and is open hemi-continuous for n ∈ Z−. • The mapping F1 is with compact value. • For all x ∈ X, ϕ(x,0) = {x}. • For all n,m ∈ Z with nm ≥ 0 and for all x ∈ X, ϕ(ϕ(x,n),m) = ϕ(x,n + m). • For all x,y ∈ X, x ∈ ϕ(y,−1) ⇔ y ∈ ϕ(x,1). c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 185 V. Milani and S. M. H. Mansourbeigi Remark 4.3. With the above notations if we let (ϕ(x,1)) := F(x), then it follows that for all x ∈ X and n ≥ 1, ϕ(x,n) = Fn(x), where Fn(x) = F(Fn−1(x)) := ⋃ {F(z) : z ∈ Fn−1(x)} is defined inductively. So F : X → X is called the generator of the discrete multivalued dynamical system. Let A be a unital C*-algebra and let Prim(A) be the topological space associated with it as in the construction of the previous sections. Define two mappings F,G : Prim(A) →P(Prim(A)) F(I) = WI = {J ∈ Prim(A) : J ⊇ I} G(I) = OI = {J ∈ Prim(A) : J ⊆ I} . Lemma 4.4. For all I,J ∈ prim(A) we have J ∈ OI ⇔ I ∈ WJ. Proof. We have J ∈ OI ⇔ J ⊆ I ⇔ I ∈ WJ. � For the mappings F,G defined above we have Proposition 4.5. The mapping F is closed hemi-continuous and the mapping G is open hemi-continuous. Proof. Let I ∈ Prim(A), W ⊆ Prim(A) be closed and F(I) = WI ⊆ W. We show that there exists a closed subset K ⊆ Prim(A) with I ∈ K and F(K) ⊆ W. Set K := WI. Then we have F(WI) = ⋃ {F(J) : J ∈ WI} = ⋃ {F(J) : J ⊇ I}⊆ WI ⊆ W. Since for J ⊇ I, we have WJ ⊆ WI. Now let I ∈ Prim(A), O ⊆ Prim(A) be open and G(I) = OI ⊆ O. We show that there exists a open subset U ⊆ Prim(A) with I ∈ U and G(U) ⊆ O. Set U := OI. Then we have G(OI) = ⋃ {G(J) : J ∈ OI} = ⋃ {G(J) : J ⊆ I}⊆ OI ⊆ O. Since for J ⊆ I, we have OJ ⊆ OI. � In the following we will see how the above mappings F,G generate a discrete multivalued dynamical system on Prim(A). Let ϕ : Prim(A) ×Z+ →P(Prim(A)) be defined in the following way: For all I ∈ Prim(A), set ϕ(I,1) := F(I) = WI. For n ∈ Z +, define ϕ(I,n) inductively by ϕ(I,n) := Fn(I) = F(Fn−1(I)) = ⋃ {F(In−1) : In−1 ∈ F n−1(I)} c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 186 Discrete dynamics on noncommutative CW complexes = ⋃ In−1⊇In−2 ... ⋃ I1⊇I F(In−1) = ⋃ In−1⊇...⊇I1⊇I F(In−1). The mapping ϕ has the following property: Lemma 4.6. For all I ∈ Prim(A) and all n,m ∈ Z+, we have ϕ(I,n + m) = ϕ(ϕ(I,n),m). Proof. We have ϕ(I,n + m) = Fn+m(I). On the other hand ϕ(ϕ(I,n),m) = ⋃ {ϕ(J,m) : J ∈ ϕ(I,n)} = ⋃ {Fm(J) : J ∈ Fn(I)} = Fm(Fn(I)) = Fm+n(I). � In the same way let ψ : Prim(A) × Z− → P(Prim(A)) be defined in the following way: For all I ∈ Prim(A), set ψ(I,−1) := G(I) = OI. For n ∈ Z +, define ψ(I,−n) inductively by ψ(I,−n) := Gn(I) = G(Gn−1(I)) = ⋃ {G(In−1) : In−1 ∈ G n−1(I)} = ⋃ In−1⊆In−2 ... ⋃ I1⊆I G(In−1) = ⋃ In−1⊆...⊆I1⊆I G(In−1). The mapping ψ has the following property: Lemma 4.7. For all I ∈ Prim(A) and all n,m ∈ Z+, we have ψ(I,−n−m) = ψ(ψ(I,−n),−m). Proof. We have ψ(I,−n−m) = Gn+m(I). On the other hand ψ(ψ(I,−n),−m) = ⋃ {ψ(J,−m) : J ∈ ψ(I,−n)} = ⋃ {Gm(J) : J ∈ Gn(I)} = Gm(Gn(I)) = Gm+n(I) . � Proposition 4.8. Let F,G,ϕ,ψ be as before. Let Θ : Prim(A) ×Z →P(Prim(A)) be defined by Θ(I,n) = ϕ(I,n) = Fn(I); Θ(I,−n) = ψ(I,−n) = Gn(I); Θ(I,0) = {I} for all I ∈ Prim(A),n ∈ Z+. Then Θ defines a discrete multivalued dynamical system on Prim(A) with generators F,G. c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 187 V. Milani and S. M. H. Mansourbeigi Proof. We have to check the properties of definition (4.2) for Θ. First of all for each I ∈ Prim(A), Θ(I,1) is compact. Moreover from proposition (4.5), the hemi-continuity property satisfies for Θ. Also • For all I ∈ Prim(A), we have Θ(I,0) = {I}. • For all n,m ∈ Z with nm ≥ 0 it follows from lemmas (4.6) and (4.7), Θ(I,n + m) = Θ(Θ(I,n),m). And eventually from the lemma (4.4), for all I,J ∈ Prim(A) we have J ∈ Θ(I,1) = F(I) = WI ⇔ I ∈ Θ(J,−1) = G(I) = OI. � Remark 4.9. With the notations of the previous proposition, if for each W ⊆ Prim(A) we define F−1(W) := {J ∈ Prim(A) : F(J) ⊆ W}, then the proof of the above proposition shows that G = F−1. For this reason sometimes we refer to F as the only generator of the system. Definition 4.10. Let A be a unital C*-algebra, Θ : Prim(A)×Z →P(Prim(A)) be a discrete multivalued dynamical system with generator F , k,m ∈ Z+ and [−k,m] be an interval in Z containing 0 ∈ Z. Let {Ii}−k≤i≤m be a sequence in Prim(A) such that ∀−k ≤ i ≤ m ; Ii+1 ∈ F(Ii). Define a map α : [−k,m] → Prim(A) by α(i) = Ii, for all −k ≤ i ≤ m. obviously α(i + 1) ∈ F(α(i)). With these notations α is called a solution for F and the sequence{Ii}−k≤i≤m is called a trajectory for F passing through α(0) = I0. With these notations: Proposition 4.11. If α : [−k,m] → Prim(A) is a solution for F, then for each i ∈ [−k,m], α(i) ∈ Fi(α(0)). Proof. We prove the statement by induction on k,m. The induction is in two parts: positive and negative parts of the interval. For k = 0,m = 1, we have α(1) ∈ F(α(0)). Now suppose α(i) ∈ Fi(α(0)), for 0 ≤ i ≤ m. We show that α(m + 1) ∈ Fm+1(α(0)). We have Fm+1(α(0)) = F(Fm(α(0))) = ⋃ {F(J) : J ∈ Fm(α(0))}. Set J = α(m). Then F(α(m)) ⊆ Fm+1(α(0)). On the other hand we have α(m + 1) ∈ F(α(m)). So α(m + 1) ∈ Fm+1(α(0)). So the induction on the positive part is completed. Now we go through the second part of the induction. The proof of this part is the same as the first part with a minor difference. We just have to note that for k = −1,m = 0, we have α(0) ∈ F(α(−1)) = Wα(−1). Consequently α(−1) ∈ Oα(0), which means α(−1) ∈ G(α(0)) = F −1(α(0)). Now if α(i) ∈ Fi(α(0)), for −k ≤ i ≤−1. We can easily see that α(−k −1) ∈ F−k−1(α(0)). � c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 188 Discrete dynamics on noncommutative CW complexes In what follows Θ : Prim(A) × Z → P(Prim(A)) is a discrete multivalued dynamical system on Prim(A) with generator F . Definition 4.12. Let α be a solution for F and {Ii}−k≤i≤m be a trajectory for F passing through α(0) = I0. The ideal I0 is called a fixed point for F if there exist W ⊆ Prim(A) such that for all n, Fn(I0) = W . Consequently for all n, α(n) ∈ W . Definition 4.13. The unstable manifolds of F at point I ∈ Prim(A) is defined by Wu(I,F) = ⋃ n≥1 Fn(I). In the same way the stable manifold of F at I is defined by W s(I,F) = ⋃ n≥1 F −n(I) = ⋃ n≥1 G n(I). Proposition 4.14. Let I,J ∈ Prim(A) and Wu(I,F) ⋂ Ws(J,F) 6= ∅. Then there exists a trajectory {Li}0≤i≤m for F from I to J, i.e. L0 = I,Lm = J. Proof. Let L ∈ Wu(I,F) ⋂ Ws(J,F). Then L ∈ Wu(I,F) = ⋃ n≥1 F n(I). So there exists n0 ≥ 1 such that L ∈ Fn0(I) = Fn0−1(F(I)) = ⋃ {Fn0−1(D) : D ∈ F(I)} So there exists L1 ∈ F(I) such that L ∈ Fn0−1(L1) = F n0−2(F(L1)) = ⋃ {Fn0−2(D) : D ∈ F(L1)} So there exists L2 ∈ F(L1) with L ∈ F n0−2(L2). Continuing in this process we obtain a sequence {L1, ...,Ln0} with the property that Li+1 ∈ F(Li) and L ∈ Fn0−i(Li) for all 1 ≤ i ≤ n0 −1. Now the sequence {L0,L1, ...,Ln0} with L0 = I,Ln0 = L is a trajectory for F from I to L. On the other hand we have L ∈ Ws(J,F) = ⋃ n≥1 F −n(J) = ⋃ n≥1 G n(J). So there exists m ≥ 1 such that L ∈ Gm(J) = Gm−1(G(J)) = ⋃ {Gm−1(D) : D ∈ G(J)} So there exists Dm−1 ∈ G(J) such that L ∈ Gm−1(Dm−1) = G m−2(G(Dm−1)) = ⋃ {Gm−2(D) : D ∈ G(Dm−1)} So there exists Dm−2 ∈ G(Dm−1) with L ∈ G m−2(Dm−2). Continuing in this process we obtain a sequence {D1, ...,Dm−1} with the property that Dm−i ∈ G(Dm−i+1), for all 2 ≤ i ≤ m− 1 and L ∈ G m−i(Dm−i), for 1 ≤ i ≤ m − 1. This means that Dm−i+1 ∈ F(Dm−i), for all 2 ≤ i ≤ m − 1. Set Dm = J. From Dm−1 ∈ G(J), it follows that J ∈ F(Dm−1). Now the sequence {D0,D1, ...,Dm−1,Dm} with D0 = L,Dm = J is a tra- jectory for F from L to J. If we rename Di = Ln0+i, for 0 ≤ i ≤ m, then the sequence {L0,L1, ...,Ln0+m} is a trajectory for F from I to J. � c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 189 V. Milani and S. M. H. Mansourbeigi In the next section we go through an example to have a better understanding of the constructions of this section. 5. Dynamical system on C(T 2) In [12] we explained how the CW complex structure for a compact topolog- ical space X induces a noncommutative CW complex structure on the algebra C(X) of continuous functions on X. In this part we apply the techniques of the previous section to introduce a discrete multivalued dynamical system on the noncommutative CW complex structure of C(T 2): the algebra of continuous functions on the 2-dimensional torus. We compute the stable and nonstable manifolds and explain the geometry of the noncommutative CW complex by its stable and unstable manifolds. Consider the following CW complex structure for the torus T 2. X0 = {0} is the one point set, X1 = {α,β}, where α,β are closed curves homeomorphic images of the circle S1, starting at point 0 and X2 = T 2. The noncommutative CW complex decomposition on C(T 2) is induced as: A0 = C(X0),A1 = C(X1),A = A2 = C(T 2). To each x ∈ X there corresponds an ideal Ix ∈ Prim(C(X)) defined by Ix := {f ∈ C(X) : f(x) = 0}. We can partition Prim(A) into three classes of ideals: There is only one 0-ideal defined by I0 := {f ∈ A : f(0) = 0}. There are two 1-ideals defined by Iα := {f ∈ A : f(x) = 0;x ∈ α} = ⋂ x∈α Ix, Iβ := {f ∈ A : f(x) = 0;x ∈ β} = ⋂ x∈β Ix. There is one 2-ideal defined by I := {f ∈ A : f(x) = 0;x ∈ T 2} = {0}. Obviously I ⊆ Iα,Iβ ⊆ I0. We have W0 = {J ∈ Prim(A) : J ⊇ I0} = {I0} WI = {J ∈ Prim(A) : J ⊇ I} = {I0,Iα,Iβ,I} Wα = {J ∈ Prim(A) : J ⊇ Iα} = {I0,Iα} Wβ = {J ∈ Prim(A) : J ⊇ Iβ} = {I0,Iβ} And W0 ⊆ Wα,Wβ ⊆ WI. On the other hand we have O0 = {J ∈ Prim(A) : J ⊆ I0} = {I0,Iα,Iβ,I} OI = {J ∈ Prim(A) : J ⊆ I} = {I} Oα = {J ∈ Prim(A) : J ⊆ Iα} = {I,Iα} Oβ = {J ∈ Prim(A) : J ⊆ Iβ} = {I,Iβ} And OI ⊆ Oα,Oβ ⊆ O0. Now we start our computations. c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 190 Discrete dynamics on noncommutative CW complexes • The ideal I0: We have Θ(I0,1) = F(I0) = W0 = {I0}. Also Θ(I0,2) = F 2(I0) = F(F(I0)) = F(W0) = ⋃ {F(J) : J ∈ W0} = F(I0) = {I0}. Continuing with this process, we see that for each n ≥ 1 we have Θ(I0,n) = F n(I0) = {I0} = W0. We have Θ(I0,−1) = G(I0) = O0 = {I0,Iα,Iβ,I} Θ(I0,−2) = G 2(I0) = G(G(I0)) = ⋃ {G(J) : J ∈ G(I0)} = ⋃ {G(J) : J = I0,Iα,Iβ,I} = G(Io) ⋃ G(Iα) ⋃ G(Iβ) ⋃ G(I) = O0 = {I0,Iα,Iβ,I} In the same way we see that for all n ≥ 1, Θ(I0,−n) = F −n(I0) = G n(I0) = {I0,Iα,Iβ,I} = O0 • The ideal Iα: We have Θ(Iα,1) = F(Iα) = Wα = {I0,Iα}. Also Θ(Iα,2) = F 2(Iα) = F(F(Iα)) = F(Wα) = ⋃ {F(J) : J ∈ Wα} = ⋃ {F(J) : J = I0,Iα} = F(I0) ⋃ F(Iα) = Wα. Continuing with this process, we see that for each n ≥ 1 we have Θ(Iα,n) = F n(Iα) = Wα. For the negative part We have Θ(Iα,−1) = G(Iα) = Oα = {Iα,I} Θ(Iα,−2) = G 2(Iα) = G(G(Iα)) = ⋃ {G(J) : J ∈ G(Iα)} = ⋃ {G(J) : J = Iα,I} = G(Iα) ⋃ G(I) = Oα = {Iα,I} In the same way we see that for all n ≥ 1, Θ(Iα,−n) = F −n(Iα) = G n(Iα) = Oα = {Iα,I} • The ideal Iβ: For this ideal as in the case of Iα we can see that for all n ∈ Z +, Θ(Iβ,n) = F n(Iβ) = {Iβ,I0} Θ(Iβ,−n) = F −n(Iβ) = {Iβ,I} • The ideal I: We have Θ(I,1) = F(I) = WI = {I0,Iα,Iβ,I}. Also Θ(I,2) = F2(I) = F(F(I)) = F(WI) = ⋃ {F(J) : J ∈ WI} = F(I0) ⋃ F(Iα) ⋃ F(Iβ) ⋃ F(I) = {I0,Iα,Iβ,I}. Continuing with this process, we see that for each n ≥ 1 we have Θ(I,n) = Fn(I) = WI = {I0,Iα,Iβ,I} We have Θ(I,−1) = G(I) = OI = {I} c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 191 V. Milani and S. M. H. Mansourbeigi Θ(I,−2) = G2(I) = G(G(I)) = ⋃ {G(J) : J ∈ G(I)} = G(I) = {I} In the same way we see that for all n ≥ 1, Θ(I,−n) = F−n(I) = Gn(I) = {I} = OI Now we explain the trajectories of the system and find the fixed points and observe the behavior of the stable and unstable manifolds at the fixed points. First we consider the sequence {I,Iα,I0}. For this sequence we have Iα ∈ F(I), I0 ∈ F(Iα) Therefore the sequence defines a trajectory for the system. Now if we define a curve σ : [0,2] ⊆ Z → Prim(A) by σ(0) = I, σ(1) = Iα, σ(2) = I0 then we have σ(n) ∈ F(σ(n − 1)) and σ(n) ∈ Fn(σ(0)) for n = 1,2. On the other hand Fn(σ(0)) = Fn(I) = WI. So I is a fixed point for this trajectory. The unstable and stable manifolds would be Wu(I,F) = ⋃ n=1,2 Fn(I) = WI = {I0,Iα,Iβ,I}. Ws(I,F) = ⋃ n≥1 F−n(I) = ⋃ n=1,2 Gn(I) = OI = {I}. From the above calculations we can conclude that the whole Prim(A) is un- stable and the ideal I corresponded to the critical chain WI of the modified discrete function on prim(A) is stable, we refer to [12] for details on critical chains. This critical chain corresponds to the maximum point of the Morse height function on T 2. Since the compact torus T 2 is homeomorphic to the space Prim(A) [7], this is a natural conclusion comparing to the unstability of torus. We have another beautiful interpretation: Wu(I0,F) = ⋃ n Fn(I0) = W0 = {I0}. Ws(I0,F) = ⋃ n≥1 F−n(I0) = ⋃ n≥1 Gn(I0) = O0 = {I0,Iα,Iβ,I}. Which means that the stable and unstable manifolds are interchanged along suitable trajectories. c© AGT, UPV, 2013 Appl. Gen. 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