TTRANSITION MATRIX THEORY
ROBERT FRANZOSA AND EWERTON R. VIEIRA Abstract.
In this article we present a unification of the theory of algebraic, singular, topological anddirectional transition matrices by introducing the (generalized) transition matrix which encompasses eachof the previous four. Some transition matrix existence results are presented as well as verification thateach of the previous transition matrices are cases of the (generalized) transition matrix. Furthermore weaddress how applications of the previous transition matrices to the Conley Index theory carry over to the(generalized) transition matrix.Dedicated to the memory of James Francis Reineck
Contents
1. Introduction 12. Transition Matrix 83. Algebraic Existence Results 104. Algebraic Transition Matrix. 125. Singular Transition Matrix 136. Topological Transition Matrix. 207. Directional Transition Matrix 23References 291.
Introduction
The Conley index theory has been a valuable topological technique for detecting global bifurcations indynamical systems [C], [CF], [FiM], [Fr1], [Fr2], [Fr3] and [MM1]. This index is a standard tool in the analysisof invariant sets in dynamical systems, and its significance owes partly to the fact that it is invariant underlocal perturbation of a flow (the continuation property). Typically, one does not investigate a single invariantset in a dynamical system but rather works with decompositions of a larger invariant set into invariant subsetsand connecting orbits between them. The Morse decomposition is the standard such decomposition in theConley index theory. Within the index theory there are matrices of maps defined between the Conley indicesof invariant sets in a Morse decomposition, and these matrices (the connection matrices) provide information
Mathematics Subject Classification.
Primary 37B30; 37D15; Secondary 70K70; 70K50; 55T05.
Key words and phrases.
Conley index, Connection Matrices, Transition Matrices, Morse-Smale System, Fast-Slow Systems. Supported by FAPESP under grant 2010/19230-8. a r X i v : . [ m a t h . D S ] N ov FRANZOSA AND VIEIRA about connections that exist between sets in the decomposition. The connection matrices also have localinvariance properties under continuation. Nevertheless, under global continuation sets of connection matricescan undergo change which usually means that the dynamical system has undergone a global bifurcation.An initial approach to identify bifurcations via the interplay between local invariance and global change inconnection matrices was due to Reineck in [R](1988). By introducing an artificial slow flow on the parameterspace in a continuous family of dynamical systems, he obtained a map between the Conley indices of Morsedecomposition invariant sets at the initial parameter value and those at the final parameter value in acontinuation. This map is known as a singular transition matrix, and it has the feature that a nonzero entrycan identify a change of the connecting orbit structure of the Morse decomposition under the continuation.Following Reineck’s work on singular transition matrices, Mischaikov and McCord in [MM1](1992) usedthe continuation property of the Conley index, without introducing an artificial slow flow, to define matricesof maps between the Conley indices of Morse decomposition invariant sets at the initial and final parametervalues in a continuation. Their maps, known as topological transition matrices, are naturally-defined mapson Conley indices that arise from the topological structure of invariant sets in the flow on the larger “phase-cross-parameter” space. In order to define these maps on the indices of the Morse decomposition sets, it wasnecessary to assume that connection matrices were trivial at the end parameter values. Nonetheless, as withthe singular transition matrices, they were able to demonstrate that non-trivial topological transition matrixentries identify potential bifurcations that exist in the overall continuation. Furthermore, in [MM2] theyestablished an equivalence between the singular transition matrices and the topological transition matricesin instances where both are defined. Later Franzosa, de Rezende and Vieira [FdRV](2014) defined a new(general) topological transition matrix that extends the previous one, not requiring the assumption that theconnection matrices are trivial at the end parameter values. In their case, the general topological transitionmatrix is defined to cover naturally-defined Conley-index maps rather than being defined directly by them.In [FM](1995), Franzosa and Mischaikov introduced the concept of an algebraic transition matrix. Giventhat connection matrices for a Morse decomposition are not unique, they raised the question of whetherthe nouniqueness could be understood via similarity transformations between connection matrices. Suchtransformations are algebraically defined, and - besides being associated with nonuniqueness of connectionmatrices for a particular Morse decomposition - can be exploited to track changes in connection matricesunder flow continuation. They developed an existence result for algebraic transition matrices under the as-sumption of a “stackable” underlying partial order, and they demonstrated how algebraic transition matricescould also be used to identify global bifurcations in a dynamical system under continuation.In [KMO] and [GKMOR], the authors developed the directional transition matrix, a transformation thatis similar in nature to both the singular transition matrix and the topological transition matrix. As inthe singular transition matrix case, a slow flow is added to the parameter space, but it is more generalthan the specific flow used in defining the singular transition matrix. The advantage to the more-generalapproach is that it allows us to detect broader families of bifurcation orbits under continuation than thosethat are detected by the singular and topological transition matrices. The directional transition matrix is atransformation between indices of Morse decomposition sets at each end of the continuation, but not simply
RANSITION MATRIX THEORY 3 from those at one end of the continuation to those at the other (as in the other types of transition matrices).Instead, it maps the indices of those sets on either end that have an outward slow-flow direction to theindices of those sets with an inward slow-flow direction. As with the classical topological transition matrix,it is assumed that on each end of the continuation there are no connecting orbits between the Morse sets,so that natural flow-defined maps can be used to define the directional transition matrix. And, as in each ofthe above cases, the authors demonstrate how non-trivial directional transition entries identify bifurcationsthat occur under continuation.While these four types of transition matrices are each defined differently and in different settings, theyhave in common that each is a Conley-index based algebraic transformation that tracks changes in indexinformation under continuation and thereby identifies global bifurcations that could occur during the con-tinuation. It is natural to expect that the theories could be unified in an overarching transition matrixtheory, and that is the main purpose of this paper. The basic idea for this general transition matrix is thatit covers natural flow-defined index isomorphisms that arise under a continuation. We discuss this aspectof the transition matrix further at the start of the next section, and then we provide our definition of thetransition matrix. First, though, we present briefly the necessary background material from Conley IndexTheory, Morse decompositions, homology index braids, connection matrices, etc. (see [C], [Fr1], [Fr2], [Fr3],[MMr] and [S]).Throughout the paper P represents a finite set with a partial order < . An interval in P is a set I ⊆ P which is such that if p, q ∈ I and p < r < q then r ∈ I . The set of intervals in < is denoted by I ( < ).An adjacent n-tuple of intervals in < is an ordered collection ( I , ..., I n ) of mutually disjoint nonemptyintervals in < satisfying: • (cid:83) ni =1 I i ∈ I ( < ); • π ∈ I j , π (cid:48) ∈ I k , j < k imply π (cid:48) ≮ π. The collection of adjacent n -tuples of intervals in < is denoted I n ( < ). An adjacent 2-tuple of intervalsis also called an adjacent pair of intervals. If < (cid:48) is either an extension of < or the restriction of < to aninterval in < , then I n ( < (cid:48) ) ⊆ I n ( < ). If ( I , J ) is an adjacent pair (2-tuple) of intervals, then I ∪ J is denoted IJ . If ( I , . . . , I n ) is called a decomposition of I . Definition 1.1.
A graded module braid over < is a collection G = G ( < ) of graded modules and maps betweenthe graded modules satisfying: (1) for each I ∈ I ( < ) , there is a graded modules G ( I ) , (2) for each ( I , J ) ∈ I ( < ) , there are maps: i ( I , IJ ) : G ( I ) → G ( IJ ) of degree 0; p ( IJ , J ) : G ( IJ ) → G ( J ) of degree 0; ∂ ( J , I ) : G ( J ) → G ( I ) of degree 1, that satisfy: • · · · → G ( I ) i → G ( IJ ) p → G ( J ) ∂ → G ( I ) → · · · is exact, • if I and J are noncomparable, then p ( JI , I ) i ( I , IJ ) = id | G ( I ) , • if ( I , J , K ) ∈ I ( < ) , then the following braid diagram commutes. FRANZOSA AND VIEIRA G ( I ) G ( IJK ) G ( K ) G ( J ) G ( IJ ) G ( JK ) G ( IJ ) G ( K ) G ( J ) G ( I ) G ( IJK ) iipp∂p ∂pi∂iiip∂ ∂∂i Assume that G and G (cid:48) are graded module braids over < . Definition 1.2. an r -degree map θ : G → G (cid:48) is a collection { θ ( I ) } I ∈ I ( < ) of module homomorphisms θ ( I ) : G ( I ) → G (cid:48) ( I ) such that the following diagram commutes for each ( I , J ) ∈ I ( < ) : · · · (cid:47) (cid:47) G k ( I ) i (cid:47) (cid:47) θ ( I ) (cid:15) (cid:15) G k ( IJ ) p (cid:47) (cid:47) θ ( IJ ) (cid:15) (cid:15) G k ( J ) ∂ λ ( J , I ) (cid:47) (cid:47) θ ( J ) (cid:15) (cid:15) G k − ( I ) (cid:47) (cid:47) θ ( I ) (cid:15) (cid:15) · · ·· · · (cid:47) (cid:47) G (cid:48) k − r ( I ) i (cid:47) (cid:47) G (cid:48) k − r ( IJ ) p (cid:47) (cid:47) G (cid:48) k − r ( J ) ∂ µ ( J , I ) (cid:47) (cid:47) G (cid:48) k − r − ( I ) (cid:47) (cid:47) · · · If, futhermore, θ ( I ) is an isomorphism for each I ∈ I ( < ) , then θ is called an r -degree isomorphism and G and G are said to be r isomorphic. Definition 1.3.
A chain complex braid over < is a collection C = C ( < ) of chain complexes and chain mapssatisfying: (1) for each I ∈ I ( < ) , there is a chain complex C ( I ) , (2) for each ( I , J ) ∈ I ( < ) , there are degree maps i ( I , IJ ) : C ( I ) → C ( IJ ) and p ( IJ , J ) : C ( IJ ) → C ( J ) which satisfy: • C ( I ) i → C ( IJ ) p → C ( J ) is weakly exact, • if I and J are noncomparable, then p ( JI , I ) i ( I , IJ ) = id | C ( I ) , • if ( I , J , K ) ∈ I ( < ) , then the following braid diagram commutes. RANSITION MATRIX THEORY 5 C ( I ) C ( IJ ) C ( J ) C ( IJK ) C ( JK ) C ( K ) ii ippppi Now assume that C and C (cid:48) are chain complex braids over < . Definition 1.4. an r -degree chain map T : C → C (cid:48) is a collection of maps T ( I ) : C ( I ) → C (cid:48) ( I ) , I ∈ I ( < ) ,such that for each ( I , J ) ∈ I ( < ) the following diagram commutes: C k ( I ) i (cid:47) (cid:47) T ( I ) (cid:15) (cid:15) C k ( IJ ) p (cid:47) (cid:47) T ( IJ ) (cid:15) (cid:15) C k ( J ) T ( J ) (cid:15) (cid:15) C (cid:48) k − r ( I ) i (cid:47) (cid:47) C (cid:48) k − r ( IJ ) p (cid:47) (cid:47) C (cid:48) k − r ( J )To simplify and to agree with previous definitions, we denote: a 0 degree map θ : G → G (cid:48) by map θ ; 0degree isomorphism θ : G → G (cid:48) by isomorphism θ ; and 0 degree chain map T : C → C (cid:48) by chain map T .Let ϕ be a continuous flow on a locally compact Hausdorff space and let S be a compact invariant setunder ϕ . A Morse decomposition of S is a collection of mutually disjoint compact invariant subsets of S , M ( S ) = { M ( p ) | π ∈ P } indexed by a finite set P , where each set M ( p ) is called a Morse set . A partial order < on P is called an admissible ordering if for x ∈ S \ (cid:83) π ∈ P M ( p ) there exists p < q such that α ( x ) ⊆ M ( p ) and ω ( x ) ⊆ M ( q ) . The flow defines an admissible ordering of M , called the flow ordering of M , denoted < F , and such that M ( π ) < F M ( π (cid:48) ) if and only if there exists a sequence of distinct elements of P : π = π , . . . , π n = π (cid:48) ,where C ( M ( π j ) , M ( π j − )), the set of connecting orbit between M ( π j ) and M ( π j − ), is nonempty for each j = 1 , . . . , n . Note that every admissible ordering of M is an extension of < F .In the Conley theory one begins with the Conley index for isolated invariant sets, i. e., S ⊆ X is an isolated invariant set if there exists a compact set N ⊆ X such that S ⊆ intN and S = Inv ( N, ϕ ) = { x ∈ N | O ( x ) ⊆ N } . The homological Conley index of S , CH ∗ ( S ) is the homology of the pointed space ( N \ L ), where ( N, L ) isan index pair for S . Setting M ( I ) = (cid:91) π ∈ I M ( π ) ∪ (cid:91) π,π (cid:48) ∈ I C ( M ( π (cid:48) ) , M ( π )) , FRANZOSA AND VIEIRA the Conley index of M ( I ), CH ∗ ( M ( I )), in short H ∗ ( I ), is well defined, since M ( I ) is an isolated invariantset for all I ∈ I ( < ).Given M ( S ), a Morse decomposition of S , the existence of an admissible ordering on M ( S ) implies thatany recurrent dynamics in S must be contained within the Morse sets, thus the dynamics off the Morsesets must be gradient-like. For this reason, Conley index theory refers to the dynamics within a Morse setas local dynamics and off the Morse sets as global dynamics. We briefly introduce the connection matrixtheory, which addresses this latter aspect. Definition 1.5.
Given G , a graded module braid over < , and C = { C ( p ) } p ∈ P , a collection of graded modules,let ∆ : (cid:76) p ∈ P C ( p ) → (cid:76) p ∈ P C ( p ) be a < -upper triangular boundary map. Then: (1) If H , the graded module braid generated by ∆ , is isomorphic to G , then ∆ is called a C − connectionmatrix of G ; (2) If, furthermore, C ( p ) is isomorphic to G ( p ) for each p ∈ P , then ∆ is called a connection matrix of G . To simplify notation, for I ∈ I ( < ) we denote (cid:76) π ∈ I C ( π ) by C ( I ), and the corresponding homology modulein H ∆ by H ( I ). In particular, the homology index braid of an admissible ordering of a Morse decomposition G = { H ∗ ( I ) } I ∈ I ( < ) is an example of a graded module braid. In this setting a < -upper triangular boundarymap ∆ : (cid:77) π ∈ P CH ∗ ( M ( π )) → (cid:77) π ∈ P CH ∗− ( M ( π ))satisfying Definition 1.5 for C ∆ = { CH ∗ ( M ( π )) } π ∈ P is called the connection matrix for a Morse decom-position . Moreover, let CM ( < ) denote the set of all connection matrices for a given ( < -ordered) Morsedecomposition M ( S ).One of the key features in Conley theory is its invariance under continuation. Since the connectionmatrices for Morse decompositions are algebraically derived from the homology Conley index braid, thisseems to indicate that connecting orbits that persist over open sets in parameter space are identified byconnection matrices. We now define Conley index continuation.Let Γ be a Hausdorff topological space, Λ a compact, locally contractible, connected metric space and X a locally compact metric space. Assume that X × Λ ⊆ Γ is a local flow and Z is a locally compact space.Let Π X : X × Λ → X and Π Λ : X × Λ → Λ be the canonical projection maps. See [S] and [Fr3].
Definition 1.6.
A parametrization of a local flow X ⊆ Γ is a homeomorphism φ : Z × Λ → X such that foreach λ ∈ Λ , φ ( Z × { λ } ) is a local flow. Let φ : Z × Λ → X be a parametrization of a local flow X . Denote the restriction φ | ( Z ×{ λ } ) by φ λ andits image by X λ . Lemma 1.1. [Salamon]
For any compact set N ⊆ X the set Λ( N ) = { λ ∈ Λ | N × λ is an isolatingneighborhood in X × λ } is open in Λ . RANSITION MATRIX THEORY 7
Definition 1.7.
The space of isolated invariant sets is S = S ( φ ) = { S × λ | λ ∈ Λ and S × λ is an isolated invariant compact set in X × λ } . For all compact sets N ⊆ X define the maps (cid:37) N : Λ( N ) → S and (cid:37) N ( λ ) = Inv ( N × λ ). Then considerthe topology on the space S generated by the sets { (cid:37) N ( U ) | N ⊆ X compact, U ⊆ Λ( N ) open } .A map γ : Λ → S is called a section of the space of isolated invariant sets if Π Λ ◦ γ = id | Λ .We are interested in the situation where the homology index braids of admissible orderings of Morsedecompositions at parameters λ and µ are isomorphic. That is, it is not enough that a Morse decompositioncontinues over Λ, it must also continue with a partial order, more specifically: Definition 1.8.
Let M ( S ) = { M ( π ) | π ∈ ( P , < ) } be an ordered Morse decomposition of the isolatedinvariant set S ⊆ X × Λ . Let M λ = { M µ ( π ) } π ∈ ( P ,< λ ) , M µ = { M µ ( π ) } π ∈ ( P ,< µ ) , S λ and S µ be the setsobtained by intersection of M ( S ) and S by the fibers X × { λ } and X × { µ } , respectively, where < ν is theorder restricted to the order < in the parameter ν ∈ Λ . • We say that M ( S ) with its order < continues over Λ if there exist sections σ and ς π : Λ → S suchthat { ς π ( ν ) | π ∈ ( P , < ν ) } is a Morse decomposition for σ ( ν ) , ∀ ν ∈ Λ . • If, furthermore, there exist a path ω : [0 , → Λ from λ to µ ; σ ( λ ) = S λ ; σ ( µ ) = S µ ; ς π ( λ ) = M λ ( π ) ; ς π ( µ ) = M µ ( π ) ; and if M ( S ) continues at least over ω ([0 , , then we say that the admissibleorderings < λ and < µ are related by continuation or continue from one to the other. See Figure 1. Λ Λ id M λ S λ M µ S µ ςσ Π Λ σ, ς Figure 1.
Sections from Definition 1.8The following Lemma 1.2 is helpful to understand the previous Definition 1.8.
Lemma 1.2. [McCord, Mischaikov, Salamon] • Let γ : Λ → S be a section, then γ is continuous if and only if S = (cid:91) λ ∈ Λ γ ( λ ) FRANZOSA AND VIEIRA is an isolated invariant set in X × Λ . • Let S = (cid:91) λ ∈ Λ σ ( λ ) , M ( π ) = (cid:91) λ ∈ Λ ς π ( λ ) for any π ∈ P . Then, S is an isolated invariant set in X × Λ under φ and M ( S ) = { M ( π ) | π ∈ ( P , < ) } is itsMorse decomposition if, and only if, M ( S ) with its order continues. Suppose that S and S are invariant sets related by continuation in X λ and X λ . Hence, there exists amap ω : [0 , → Λ such that ω (0) = λ and ω (1) = λ and an isolated invariant set S over ω ( I ) such that S λ i = S i . The inclusion f i : X λ i → X × ω ( I ) induces an isomorphism CH ∗ ( S i ) f i ∗ −→ CH ∗ ( S ), where CH ∗ ( S i )and CH ∗ ( S ) indicates the Conley homology indices of S i in X λ i and of S in X × ω ( I ), respectively. Thus,there is an isomorphism, called Conley flow-define isomorphism F ω : CH ∗ ( S ) f − ∗ ◦ f ∗ −→ CH ∗ ( S ) , that depends on the endpoint-preserving homotopy class ω . If π (Λ) = 0 then F ω is independent of the path ω and one writes F λ ,λ instead of F ω . The Conley flow-defined isomorphism is well-behaved with respectto composition of paths: F λ,λ = id , F µ,ν ◦ F λ,µ = F λ,ν and F λ,µ = F − µ,λ . For more details see [MM2] and [S].To simplify notation, we denote CH ∗ ( M ν ( I )) = H ∗ ,ν ( I ) or just CH ( M ν ( I )) = H ν ( I ), where I ∈ I ( < ν ) and ν ∈ { λ, µ } . 2. Transition Matrix
Given a Morse decomposition of an isolated invariant set S in a flow, an associated homology index braidcontains a significant amount of algebraic information about the overall structure of the Morse sets andconnecting orbits in S . Generally speaking, the homology index braid is a relatively unmanagable structureto examine and analyze in order to draw conclusions about the structure of S . Overcoming this difficultyis part of the motiviation for the connection matrix. The connection matrix is a matrix of maps betweenthe indices of the Morse sets that produces a graded module braid that is isomorphic to the homologyindex braid. In this way we can think of the connection matrix as “covering” the homology index braid.Consequently, it is reasonable to expect that the connection matrix itself contains information from whichthe invariant set structure can be understood.The transition matrices discussed in the introduction can be viewed similarly, and this perspective moti-vates our general transition matrix defined below. Given an invariant set and ordered Morse decompositionthat continues over a parameter interval, there is a family of continuation-defined isomorphisms betweenindices of the invariant subsets that continue. If bifurcations occur during the continuation, then this willbe reflected in a change in the flow-defined mappings in the homology index braids on either end of the con-tinuation. The idea behind the general transition matrix is that it is a matrix of mappings between indicesof Morse sets from either end of the continuation that “covers” the continuation-defined isomorphisms. Bycovering the continuation-defined isomorphisms, the transition matrix then assists in detecting change that RANSITION MATRIX THEORY 9 occurs in the homology index braids, and thus it is reasonable to expect that the transition matrix containsinformation revealing change in the invariant set structure under continuation.In this section we introduce our general definition of the transition matrix, and we address some straight-forward properties of it. It is important to have the viewpoint that the transition matrix theory that wepresent here is an algebraic theory. In fact, the same could be said for the various previous versions oftransition matrices. The primary motivation for introducing and studying transition matrices has been inapplications to the Conley index theory, but there are important questions that need to be addressed regard-ing algebraic aspects of transition matrices. In section 3 we show how previous transition matrix existenceresults can be adapted to provide specific existence results for the transition matrix introduced here. Insections 4-7 we examine applications of the transition matrix in the Conley index theory, demonstrating howprevious results using specific versions of the transition matrix carry over to our general setting.
Definition 2.1.
Given chain complex braids C and C (cid:48) and graded module braids G and G (cid:48) , an r -degree chainmap T : C → C (cid:48) is said to cover an r -degree isomorphism θ (relative to Φ and Φ (cid:48) ) if for all I ∈ I ( < ) , wehave that the following diagram commutes HC ( I ) T ∗ ( I ) (cid:47) (cid:47) Φ( I ) (cid:15) (cid:15) HC (cid:48) ( I ) Φ (cid:48) ( I ) (cid:15) (cid:15) G ( I ) θ ( I ) (cid:47) (cid:47) G (cid:48) ( I ) where T ∗ ( I ) is the homology map induced from the chain map T ( I ) , and Φ :
HC → G and Φ (cid:48) : HC (cid:48) → G (cid:48) aregraded module braid homology isomorphisms. Definition 2.2.
If, in Definition , C and C (cid:48) arise from connection matrices ∆ : (cid:76) P C ( p ) → (cid:76) P C ( p ) , ∆ (cid:48) : (cid:76) P C (cid:48) ( p ) → (cid:76) P C (cid:48) ( p ) , respectively, and T arises from a matrix T : (cid:76) P C ( p ) → (cid:76) P C (cid:48) ( p ) then T is calledan r -degree (generalized) transition matrix for ∆ and ∆ (cid:48) . Beyond this point in the paper, “transition matrix” refers to the transition matrix as defined here unless wespecifically refer to a particular previous type such as “algebraic transition matrix”, “topological transitionmatrix”, etc. Also we denote a 0 degree transition matrix T by transition matrix T . Theorem 2.1.
Let ∆ : (cid:76) P C ( p ) → (cid:76) P C ( p ) and ∆ (cid:48) : (cid:76) P C (cid:48) ( p ) → (cid:76) P C (cid:48) ( p ) be connection matrices of G and G (cid:48) , respectively. Let T : (cid:76) P C ( p ) → (cid:76) P C (cid:48) ( p ) be a transition matrix for ∆ and ∆ (cid:48) . Assume that Φ :
HC → G and Φ (cid:48) : HC (cid:48) → G (cid:48) are graded module braid isomorphisms, and that T covers an isomorphism θ (relative to Φ and Φ (cid:48) ). Then the transition matrix T satisfies the following properties: (i): T ◦ ∆ = ∆ (cid:48) ◦ T ; (ii): T ( { p } ) = id and T is upper triangular with respect to < ; (iii): T is an isomorphism; (iv): T − covers θ − (relative to Φ (cid:48) and Φ ). Moreover, suppose that T and T cover θ : G → G (cid:48) (relative to Φ and Φ (cid:48) ) and θ : G (cid:48) → G (cid:48)(cid:48) (relative to Φ (cid:48) and Φ (cid:48)(cid:48) ), and arise from T and T ,respectively. Then T ◦ T covers θ ◦ θ (relative to Φ and Φ (cid:48)(cid:48) ) and arises from T ◦ T . Proof: ( i ) Since ∆( I ) and ∆ (cid:48) ( I ) are boundary maps and T ( I ) is a chain map, we have that T ( I ) ◦ ∆( I ) = ∆ (cid:48) ( I ) ◦ T ( I )for all I .( ii ) Change F for θ in the proof of Theorem 5 in [FdRV].( iii )By item (ii) we have that T is the identity on the diagonal and upper triangular, thus T is anisomorphism.( iv ) It follows directly by the composition of the maps. (cid:3) Algebraic Existence Results
As indicated in the previous section, the transition matrix theory is an algebraic theory, and there areimportant questions to be addressed regarding the theory solely from the algebraic viewpoint. The mostbasic is the existence question: Given connection matrices ∆ and ∆ (cid:48) , does there exist a transition matrix forthem? The answer is unknown for this most general situation, however, the previous existence results for thespecific types of transition matrices provide us with positive existence results under appropriate conditions.In this section, we present a few such results.To begin, we consider the situation that is the algebraic basis behind the existence of topological transitionmatrices in [MM1]. In this case we have trivial connection matrices ∆ : (cid:76) P C ( p ) → (cid:76) P C ( p ) and ∆ (cid:48) : (cid:76) P C (cid:48) ( p ) → (cid:76) P C (cid:48) ( p ). So, for each I ∈ I ( < ) we have HC ( I ) = (cid:76) I C ( p ) and HC (cid:48) ( I ) = (cid:76) I C (cid:48) ( p ). Supposethat the associated graded module briad isomorphisms are Φ : HC → G and Φ (cid:48) : HC (cid:48) → G (cid:48) . Then for each I ∈ I ( < ), the corresponding isomorphisms are mappings Φ( I ) : (cid:76) I C ( p ) → G ( I ) and Φ (cid:48) ( I ) : (cid:76) I C (cid:48) ( p ) →G (cid:48) ( I ). We have the following, straighforward, transition matrix existence result for this case. The proof isstraightforward. Theorem 3.1.
Let ∆ and ∆ (cid:48) be connection matrices of G and G (cid:48) , respectively. Assume that ∆ and ∆ (cid:48) aretrivial and that Φ :
HC → G and Φ (cid:48) : HC (cid:48) → G (cid:48) are graded module braid isomorphisms. If Θ :
G → G (cid:48) is anisomorphism between graded module braids, then T = Φ − ( P ) ◦ Θ( P ) ◦ Φ( P ) : (cid:76) P C ( p ) → (cid:76) P C (cid:48) ( p ) is atransition matrix for ∆ and ∆ (cid:48) . Next we address an existence result that is based on the algebraic transition matrix existence result in[FM]. In this case the result holds where the underlying partial order is of a particular type. Let ( I , ..., I n )be an adjacent n-tuple of intervals whose union is I , then ( I , ..., I n ) is called a decomposition of I , if I , J ⊆ P are disjoint then we say that I and J are noncomparable if neither p < q nor p < q for every p ∈ I , q ∈ J and we say that J is totally greater than I if p < q for every p ∈ I and q ∈ J . A partial order < on P iscalled stackable if there is a decomposition of P , ( I , ..., I n ), such that < restricted to each I i is trivial andsuch that if i < j then I j is totally greater than I i . Note that a trivial order is stackable and a total orderis stackable. Definition 3.1.
Given chain complex braids C and C (cid:48) and graded module braids G and G (cid:48) , a chain map T : C → C (cid:48) is said to weakly cover an isomorphism θ (relative to Φ and Φ (cid:48) ) if there exists I ∈ I ( < ) such RANSITION MATRIX THEORY 11 that the following diagram commutes HC ( I ) T ∗ ( I ) (cid:47) (cid:47) Φ( I ) (cid:15) (cid:15) HC (cid:48) ( I ) Φ (cid:48) ( I ) (cid:15) (cid:15) G ( I ) θ ( I ) (cid:47) (cid:47) G (cid:48) ( I ) where T ∗ ( I ) is the homology map induced by the chain map T ( I ) , Φ :
HC → G and Φ (cid:48) : HC (cid:48) → G (cid:48) areisomorphisms from the homology the graded module braid to graded module braid. Note that the internal I , in the previous definition, may be P . Furthermore, we could have a collection ofinterval I ⊆ I ( < ) such that for all I ∈ I the previous diagram commutes, in this case, we say that T weaklycovers θ on I . Theorem 3.2.
Let ∆ : (cid:76) P C ( p ) → (cid:76) P C ( p ) and ∆ (cid:48) : (cid:76) P C (cid:48) ( p ) → (cid:76) P C (cid:48) ( p ) be connection matrices of G and G (cid:48) , respectively, and let θ : G → G (cid:48) be an isomorphism. If the order < is stackable with a decomposition I = { I , ..., I n } , then there is a transition matrix T : (cid:76) P C ( p ) → (cid:76) P C (cid:48) ( p ) which weakly cover θ on I . Proof:
It follows from the proof of theorem 3.5 in [FM], by checking that the induction process of applyingTheorem 3.8 in [FM] gives an collection of interval that equals to I . (cid:3) Note that trivial order and total order are stackable. Moreover, in [FM] is given a definition of N-freeorder that helps to check if the order is stackable, in general, it is more likely a partial order be a stackableorder than the opposite. However if < is not stackable one still can analyze any interval I by applyingTheorem 3.2 in order to obtain a transition matrix T ( I ) that covers θ ( I ), in other words, it is possible to getinformation between p, q ∈ P by considering any interval which has p and q .Now we explore a special order < k that leads us to another existence result by using a similar idea comingfrom the Morse index of gradient-like flows. But before defining < k , we present the following lemma. Lemma 3.1.
If there is at most one k ( p ) such that C k ( p ) ( p ) (cid:54) = 0 , for all p ∈ P , then ∆( p, p (cid:48) ) = 0 for all p, p (cid:48) ∈ P such that | k ( p ) − k ( p (cid:48) ) | (cid:54) = 1 or p (cid:48) < p . Proof:
Since ∆ is a boundary map and C ( p ) = C ( p (cid:48) ) = 0 except in dimension k ( p ) and k ( p (cid:48) ), we obtainthat ∆( p, p (cid:48) ) = 0 when | k ( p ) − k ( p (cid:48) ) | (cid:54) = 1. If p (cid:48) < p , it follows by upper triangularity of ∆ that ∆( p, p (cid:48) ) = 0. (cid:3) Under the hypothesis of Lemma 3.1, define < k a partial order in the set P such that p < k p (cid:48) ⇔ k ( p ) < k ( p (cid:48) ) . The previous lemma seems to be quite obvious but plays the main role in the proof of the next theorem.
Theorem 3.3.
Let ∆ : (cid:76) P C ( p ) → (cid:76) P C ( p ) and ∆ (cid:48) : (cid:76) P C (cid:48) ( p ) → (cid:76) P C (cid:48) ( p ) be connection matrices of G ( < k ) and G (cid:48) ( < k ) , respectively, and let θ : G → G (cid:48) be an isomorphism between graded module braids. If thereis at most one k ( p ) such that C k ( p ) ( p ) (cid:54) = 0 (cid:54) = C (cid:48) k ( p ) ( p ) , for all p ∈ P , then there exists an unique transitionmatrix T : (cid:76) P C ( p ) → (cid:76) P C (cid:48) ( p ) which covers θ . Proof:
Lemma 3.1 and the partial order < k guarantee that T is a chain map. Moreover it allows us to usethe proof of Theorem 6 in [FdRV] to obtain the result. (cid:3) Note that T obtained in the previous theorem is a matrix in block form as in Theorem 6 in [FdRV].Furthermore, it is possible to consider a generic partial order < instead of < k in Theorem 3.3, neverthelessthe proof split in various cases where < failed to be < k . However, in case < , be aware that T may not beunique neither a matrix in block form.Observe that Theorem 3.3 can be applied in a more general context than Theorem 6 in [FdRV]. Forinstance, the flow on parameters λ and µ do not need to be Morse-Smale without periodic orbits; we justneed that the Morse decomposition has to meet the hypothesis that there is at most one k ( p ) such that CH k ( p ) ( M ( p )) (cid:54) = 0, for all p ∈ P . Such a property is not satisfied for some Morse sets such as repellerperiodic orbits and attractor periodic orbits, however saddle-saddle connections with same index satisfiesthis hypothesis. 4. Algebraic Transition Matrix.
The algebraic transition matrix theory is developed in [FM]. There the authors define the algebraictransition matrix in the setting of a parameterized family of flows after having earlier developed an algebraictheory that they employ. Their algebraic transition matrix is an example of a similarity transformationdefined between connection matrices.Let ∆ : (cid:76) P C ( p ) → (cid:76) C ( p ) be a connection matrix for a graded module braid G over a partial order < .If T : (cid:76) C ( p ) → (cid:76) C (cid:48) ( p ) is < -upper triangular matrix such that T ( p ) is an isomorphism for all p ∈ P , thenit follows that ∆ (cid:48) := T ∆ T − is also a connection matrix for G . In this way (as referred to above) T can bethought of as a similarity transformation between connection matrices ∆ and ∆ (cid:48) . Proposition 4.1. [Franzosa]
Let C = { C ( p ) } p ∈ P and C (cid:48) = { C (cid:48) ( p ) } p ∈ P be collections of graded modules,and ∆ : (cid:76) P C ( p ) → (cid:76) P C ( p ) and ∆ (cid:48) : (cid:76) P C (cid:48) ( p ) → (cid:76) P C (cid:48) ( p ) be < -upper triangular boundary maps. If T : (cid:76) P C ( p ) → (cid:76) P C (cid:48) ( p ) is < -upper triangular and such that T ∆ = ∆ (cid:48) T , then { T ( I ) } I ∈ I ( < ) is a chain mapfrom C ∆ to C ∆ (cid:48) . Here, for the purpose of unifying the various transition matrix theories–and being most general–we definethe above-mentioned similarity transformation between connection matrices as follows:
Definition 4.1.
Let ∆ : (cid:76) P C ( p ) → (cid:76) P C ( p ) and ∆ (cid:48) : (cid:76) P C (cid:48) ( p ) → (cid:76) P C (cid:48) ( p ) be connection matrices. If T : (cid:76) P C ( p ) → (cid:76) P C (cid:48) ( p ) is < -upper triangular such that T ( p ) is an isomorphism for all p ∈ P and suchthat ∆ (cid:48) T = T ∆ , then T is called an algebraic transition matrix from ∆ to ∆ (cid:48) . The algebraic transition matrix as defined in [FM] then is an example of an algebraic transition matrixin Definition 4.1 above. With the current definition of algebraic transition matrix, we then show that it is atransition matrix.Note that, given ∆, ∆ (cid:48) and T as above, T is a chain map between C ∆ and C ∆ (cid:48) (see Proposition 4.1).Furthermore T induces homology isomorphisms T ∗ ( I ) : H ∆( I ) → H ∆ (cid:48) ( I ) for all I ∈ I ( < ). Now, if ∆ is aconnection matrix and Φ : H ∆ → G is a graded module braid isomorphism associated with ∆, then if for RANSITION MATRIX THEORY 13 each I ∈ I ( < ) we define Φ (cid:48) ( I ) = Φ( I ) ◦ T − ∗ ( I ), then the isomorphisms Φ (cid:48) ( I ) define a graded module braidisomorphism Φ (cid:48) between H ∆ (cid:48) and G . Also, clearly, for each I ∈ I ( < ), the following diagram commutes H ∆( I ) T ∗ ( I ) (cid:47) (cid:47) Φ( I ) (cid:15) (cid:15) H ∆ (cid:48) ( I ) Φ (cid:48) ( I ) (cid:15) (cid:15) G ( I ) id ( I ) (cid:47) (cid:47) G ( I )where id ( I ) is the identity map on G ( I ). Thus T covers id : G → G (relative to Φ and Φ (cid:48) ), and it follows that T is a transition matrix.It follows from above, that given a connection matrix ∆ for a graded module braid G , then other connectionmatrices for G can be obtained via transition matrices. The converse question is significant; that is, givenconnection matrices, ∆ and ∆ (cid:48) , is there a transition matrix for them? This question was addressed withsome initial positive results in [FM] and in the previous section 3.Define ATM( < ) to be the set of all algebraic transition matrices which are similarity transformationsbetween connection matrices ∆ : (cid:76) ( P ,< ) C ( p ) → (cid:76) C ( p ) and ∆ (cid:48) : (cid:76) ( P ,< ) C (cid:48) ( p ) → (cid:76) C (cid:48) ( p ). From theexistence theorem in section 3, one can prove the next theorem by applying the same idea from Theorem4.3 in [FM] and Lemma 3 in [FdRV]. Theorem 4.1.
Let M λ = { M λ ( π ) } π ∈ ( P ,< λ ) and M µ = { M µ ( π ) } π ∈ ( P ,< µ ) be Morse decompositions, ∆ λ and ∆ µ their respective connection matrices. Moreover, assume that M λ and M µ are related by continuation withan admissible ordering < . If < is stackable or < k and T λ,µ ( I j − , I j ) (cid:54) = 0 for all T λ,µ ∈ ATM ( < ) then thereexists s ∈ [0 , such that C (cid:0) M ω ( s ) ( I j ) , M ω ( s ) ( I j − ) (cid:1) (cid:54) = ∅ , where ω : [0 , → Λ is a path that continues M λ and M µ . (cid:3) Notice in the previous theorem, when it is possible to set I j and I j − to have only one element, Theorem4.1 will appear more closely to the next theorems about bifurcation connections between Morse sets. Ofcourse, the lack of covering the flow-defined continuation isomorphism F imposes less features for algebraictransition matrices, since covering F is intrinsically related how those connections occur along a path on Λas well as substantial existence results (see next sections).5. Singular Transition Matrix
In most cases singular transition matrices can only be computed via the dynamics of the slow system,since these transition matrices are essentially just submatrices of a connection matrix, see [R]. Thus, inpractice, one may find difficulty in obtaining those matrices if the objective is to understand the dynamicsof the parametrized family. However, by showing that those matrices are transition matrices that coveran isomorphism, we actually can use the singular transition theory to assist in the development of othertransition matrices, as we will show in the following sections.
Let(1) ˙ x = f ( x, λ ) , be a parametrized family of ordinary differential equations defined in R n , where the parameter space Λ = R .Assume that the Morse decomposition M ( S λ ) = { M λ ( p ) ∈ P } continues over R and that connection matrices∆ − and ∆ for the Morse decompositions M ( S − ) and M ( S ), respectively, are known. Moreover, let N ⊆ R n be an isolating neighborhood for S λ , λ ∈ R .One introduces slow dynamics in the parameter space of (1) with the purpose of comprehending thebifurcations that occur for − < λ <
1. Hence (1) can be written as:(2) ˙ x = f ( x, λ ) , ˙ λ = (cid:15) ( λ − (cid:15) >
0. Define M ( p + ) := M ( p ) , M ( p − ) := M − ( p )and M ( p ± ) := M ( p + ) ∪ M ( p − ) ⊆ R n × {± } . For (cid:15) > N × [ − ,
2] is an isolatingneighborhood for the flow φ (cid:15) generated by (2). Let K (cid:15) := Inv ( N × [ − , , φ (cid:15) ) . Now observe that since ˙ λ < λ ∈ ( − , (cid:15) > M ( K (cid:15) ) = { M ( p ± ) | p ∈ P } is a Morse decomposition, and there is an admissible ordering given by q − < p + ,q − < p − ⇔ q < − p,q + < p + ⇔ q < p, where < − and < are admissible orderings for M ( S − ) and M ( S ), respectively. Denoting a connectionmatrix for M ( K (cid:15) ) by ∆ (cid:15) , since the dynamics on the subspaces R n × {± } are given exactly by the flowsgenerated by ˙ x = f ( x, ± (cid:15) : (cid:77) p ∈ P H ∗ (cid:0) M ( p − ) (cid:1) (cid:77) p ∈ P H ∗ (cid:0) M ( p + ) (cid:1) → (cid:77) p ∈ P H ∗− (cid:0) M ( p − ) (cid:1) (cid:77) p ∈ P H ∗− (cid:0) M ( p + ) (cid:1) takes the form(3) ∆ (cid:15) = ∆ − T (cid:15) + where ∆ − is the connection matrix for M ( S − ) and ∆ + is the conjugation of degree 1 of the connection matrixof M ( S + ) (this conjugation is necessary because on λ = 1 we have an increase by one in the dimension of RANSITION MATRIX THEORY 15 the unstable manifold, which introduces a suspension of the Conley index). A contribution of Reineck in [R]was to formalize the expression (3). When (cid:15) → T (cid:15) is well defined and the resulting matrices arereferred to as the R-singular transition matrix . Ignoring the +1 conjugation on ∆ , it follows. Theorem 5.1. [Reineck]
An R-singular transition matrix T from λ = 1 to λ = − satisfies the followingproperties: (i): ∆ − T + T ∆ = 0 ; (ii): T is an isormorphism; (iii): T is an upper triangular matrix with respect to < ; (iv): If T ( p, q ) : H ( M ( p )) → H ( M ( q )) is nonzero, then there exists a finite sequence ≥ λ ≥ λ ≥ . . . ≥ λ k ≥ and corresponding p i ∈ P such that p i > λ i p i +1 where > λ i is the flow defined orderunder φ λ i . Now we define the singular transition matrix presented in [MM2]. As one can note, in [R] and [MM2],different suspension isomorphisms to define singular transition matrix. This difference is important to notebecause as we point at in Remark 5.1 below, those matrices are transition matrices that cover differentisomorphisms.Following the same idea as in Reineck’s development, Mischaikov and McCord in [MM2] created a newparameter space that incorporates the drift flows and the one-parameter families in Λ. More specifically, let D + = P (Λ) × G + is a parameter space for flows on X × [ − , P (Λ) = { α : [0 , → R } is the set ofpaths in Λ and G + = { g : [ − , → R | g ∈ C , g (( − , ∪ (1 , > > g ((0 , , g ( −
1) = g (0) = g (1) = g (2) = 0 , or g ≡ } . The choice of [ − ,
2] instead of [0 ,
1] is convenient, since for the drift flow ˙ s = g ( s ) on [0 ,
1] has 0 as ahyperbolic attractor and 1 as a hyperbolic repeller, therefore agreeing with the drift flow in [R]. Note thatthe domain of any path in P is [0 , τ ( s ) = − s for − ≤ s ≤ ,s for 0 ≤ s ≤ , − s for 1 ≤ s ≤ . For any ( α, g ) ∈ D + consider a flow on X × [ − ,
2] over ( α, g ) given by˙ x = f ( x, ατ ( s )) , ˙ s = g ( s ) . Observe that if α is a constant path λ , thus we obtain a product flow˙ x = f ( x, λ ) , ˙ s = g ( s ) . When || g || →
0, we have the original parameterized family of flows˙ x = f ( x, ατ ( s ))restricted to the image of α .We are interested in studying the behavior on D + ([0 , X × [0 ,
1] over ( α, g ) is therestriction of the flow on X × Λ to the one-parameter family of flows picked out by α . Let C ⊆ [0 ,
1] bea connected isolated set of zeros for g ∈ G + restricted to ( − , S is an isolated invariant set thatcontinues over α [0 ,
1] for some α then S α = { ( x, s ) | s ∈ C, x ∈ S α ( s ) } is an isolated invariant set for the flowover ( α, g ). See [MM2].Let S α ( C ) denote the restriction of S α to C ⊆ [0 , D + . Proposition 5.1. [Mischaikow-McCord] If C is a connected, isolated set of zeros of g , then h ( C ) , thehomotopy Conley index of C in [ − , , is either Σ , Σ or ¯0 . Let C g denote the component of G + thatcontains g . If S is an isolated invariant set that continues over Λ then, over P × C g , the Conley index of S α ( C ) in X × [ − , is h ( S ) ∧ h ( C ) . By the previous proposition, we have that the homology Conley index of S α ( C ) is the tensor product CH ( S α ( C )) = CH ( S ) ⊗ CH ( C ) . In this setting, we present the index suspension isomorphism Σ( S ) defined in [MM2] by the following com-position CH k ( S α ( c ) ) ⊗ σ n −−−→ CH k ( S α ( c ) ) ⊗ CH n ( C ) × −→ CH n + k ( S α ( c ) × C ) = CH n + k ( S ¯ α ( c ) ( C )) F α, ¯ α ( c ) −−−−→ CH k + n ( S α ( C )) , where c ∈ C , α is a path, ¯ α ( c ) is the constant path α ( c ) , σ n is the generator of CH n ( C ) and F α, ¯ α ( c ) is thecontinuation isomorphism along a path in D + from (¯ α ( c ) , g ) to ( α, g ). More concisely, we haveΣ( S ) : CH k ( S α ( c ) ) F α, ¯ α ( c ) ◦×◦⊗ σ n −−−−−−−−−−→ CH k + n ( S α ( C )) . Given that M ( I ) is an isolated invariant set which also continues, we can defineΣ( I ) = Σ( M ( I )) : CH k ( M ( I ) α ( c ) ) F α, ¯ α ( c ) ◦×◦⊗ σ n −−−−−−−−−−→ CH k + n ( M ( I ) α ( C )) . From the dynamics in [0 , S ( α,g ) has S α (0) as an attractor and S α (1) as a repeller. Thesecharacteristics will be the same for all g ∈ G + , even though the structure of of the connecting orbit set may RANSITION MATRIX THEORY 17 vary with g . Computing the connection matrix for the flow in X × α [0 , g : (cid:77) p ∈ P H ∗ (cid:0) M ( p ) α (0) (cid:1) (cid:77) p ∈ P H ∗ (cid:0) M ( p ) α (1) (cid:1) → (cid:77) p ∈ P H ∗ (cid:0) M ( p ) α (0) (cid:1) (cid:77) p ∈ P H ∗ (cid:0) M ( p ) α (1) (cid:1) takes the form(4) ∆ g = ∆ α (0) T g Σ α (1) where ∆ α (0) is a connection matrix for M ( S ) α (0) and ∆ Σ α (1) is the conjugation by Σ of a connection matrixof M ( S ) α (1) , see Lemma 5.1.When || g || → T g gives us a MM-singular transition matrix T s . Observe that ∆ g maydiffer from ∆ (cid:15) since we used different suspension isomorphisms to define these singular transition matrices.The following theorem points at an important difference between the structure of ∆ g and the structureof ∆ (cid:15) . Theorem 5.2. [McCord-Mischaikow]
The connection homomorphism for the attractor-repeller decomposi-tion (cid:0) M α (0) ( I ) , M α (1) ( I ) (cid:1) of M ( α,g ) ( I ) is an isomorphism, that is computed by continuation of M ( α,g ) ( I ) across Λ . That is, there is a commutative diagram H k,α (1) ( I ) id (cid:15) (cid:15) Σ( I ) (cid:47) (cid:47) H k +1 ,α (1) ( I ) δ (cid:47) (cid:47) H k,α (0) ( I ) id (cid:15) (cid:15) H k,α (1) ( I ) F α (1) ,α (0) ( I ) (cid:47) (cid:47) H k,α (0) ( I )As a consequence we have that the flow defined map of attractor-repeller map δ is an isomorphism. Thefollowing theorem addresses both the R-singular and the MM-singular transition matrix. Theorem 5.3.
A singular transition matrix is a 1-degree transition matrix.
Remark 5.1.
As we can see in the following proof, Theorem 5.4 works for both R-singular and the MM-singular transition matrix. More specifically, they are 1-degree transition matrices that cover the 1-degreeisomorphisms δ Ψ and F ◦ Σ − , respectively. Proof:
In order to simplify notation, we define λ = α (0) and µ = α (1). Let T s and T s be the R-singular andMM-singular transition matrices (defined by Ψ and Σ), respectively. Choose an interval I ∈ I ( < ), where < is the flow defined order for the product flow. Hence ( I µ , I λ ) is an attractor-repeller pair and, by ∆ g beinga connection matrix, we have that the following diagram commutes H k +1 ∆ Σ λ ( I ) Φ Σ λ ( I ) (cid:15) (cid:15) [ T s ] (cid:47) (cid:47) H k ∆ µ ( I ) Φ µ ( I ) (cid:15) (cid:15) H k +1 ,λ ( I ) δ (cid:47) (cid:47) H k,µ ( I ) . Therefore T s is a 1-degree transition matrix that covers δ = F ◦ Σ − since T s is a 1-degree chain mapand δ is defined by a 1-degree graded module braid isomorphism. If one changes Σ for Ψ we obtain T s is a1-degree transition matrix covering δ Ψ . Note that in this case we may not have δ Ψ = F ◦ Ψ − since δ Ψ is aflow defined map of the attractor-repeller pair for which the Conley index of the repeller was suspended byΨ. (cid:3) Lemma 5.1.
Σ : H λ, ∗ → H λ, ∗ +1 is a braid isomorphism between Conley index braids and ∆ Σ λ is Σ conjugatedto a connection matrix ∆ λ at parameter λ , in other words the following diagram commutes H k ∆ λ ( I ) Φ λ ( I ) (cid:15) (cid:15) [ ⊕ π ∈ I Σ( π )] (cid:47) (cid:47) H k +1 ∆ Σ λ ( I ) Φ Σ λ ( I ) (cid:15) (cid:15) H k,λ ( I ) Σ( I ) (cid:47) (cid:47) H k +1 ,λ ( I ) Proof:
Σ is an isomorphism by definition. Now it reminds to prove that for all adjacent pair ( I , J ) the followingdiagram commutes · · · (cid:47) (cid:47) H λ, ∗ ( I ) (cid:47) (cid:47) Σ( I ) (cid:15) (cid:15) H λ, ∗ ( IJ ) (cid:47) (cid:47) Σ( IJ ) (cid:15) (cid:15) H λ, ∗ ( J ) (cid:47) (cid:47) Σ( J ) (cid:15) (cid:15) · · ·· · · (cid:47) (cid:47) H λ, ∗ +1 ( I ) (cid:47) (cid:47) H λ, ∗ +1 ( IJ ) (cid:47) (cid:47) H λ, ∗ +1 ( J ) (cid:47) (cid:47) · · · Indeed, the commutativity follows from the same property that the flow-defined Conley index isomorphism F has. In other words, Σ is defined via F , thus one obtains following diagram · · · (cid:47) (cid:47) H λ, ∗ ( I ) (cid:47) (cid:47) ×◦⊗ σ ( I ) (cid:15) (cid:15) H λ, ∗ ( IJ ) (cid:47) (cid:47) ×◦⊗ σ ( IJ ) (cid:15) (cid:15) H λ, ∗ ( J ) (cid:47) (cid:47) ×◦⊗ σ ( J ) (cid:15) (cid:15) · · ·· · · (cid:47) (cid:47) H λ, ∗ +1 ( I ) (cid:47) (cid:47) F ( I ) (cid:15) (cid:15) H λ, ∗ +1 ( IJ ) (cid:47) (cid:47) F ( IJ ) (cid:15) (cid:15) H λ, ∗ +1 ( J ) (cid:47) (cid:47) F ( J ) (cid:15) (cid:15) · · ·· · · (cid:47) (cid:47) H λ, ∗ +1 ( I ) (cid:47) (cid:47) H λ, ∗ +1 ( IJ ) (cid:47) (cid:47) H λ, ∗ +1 ( J ) (cid:47) (cid:47) · · · which commutes since × ◦ ⊗ σ and F are isomorphism between braids. Therefore Σ is a graded module braidisomorphism.Note that, in order to make a cleaned proof, we omitted some subscripts which come from the definitionof Σ. But be aware that for a small enough neighborhood the Conley index H λ, ∗ ( I ) was suspended by × ◦ ⊗ σ for a constant path and soon after the suspended Conley index H λ, ∗ +1 ( I ) was homotoped via F to a path α inside a small enough neighborhood.Defining ∆ λ ( π (cid:48) , π ) := Σ − ( π ) ◦ ∆ Σ ( π (cid:48) , π ) ◦ Σ( π (cid:48) ) RANSITION MATRIX THEORY 19 we obtain that ∆ λ is a connection matrix at parameter λ since Σ : H λ, ∗ → H λ, ∗ +1 is an isomorphism betweenConley index braids, therefore the diagram in Lemma 5.1 commutes. (cid:3) MM-Singular transition matrix T s defined via Σ has another feature as follows by the next theorem. Theorem 5.4.
The MM-singular transition matrix T s composed with the induced map of (cid:76) π ∈ P Σ( π ) is atransition matrix T = T s ◦ (cid:32)(cid:77) π ∈ P Σ( π ) (cid:33) that covers Conley flow-defined isomorphim F . Proof:
Suppose that the following diagram commutes for every interval I ∈ I ( < ) H k ∆ λ ( I ) Φ λ ( I ) (cid:15) (cid:15) [ ⊕ p ∈ I Σ p ] (cid:47) (cid:47) H k +1 ∆ Σ λ ( I ) Φ Σ λ ( I ) (cid:15) (cid:15) [ T s ] (cid:47) (cid:47) H k ∆ µ ( I ) Φ µ ( I ) (cid:15) (cid:15) H k,λ ( I ) id (cid:15) (cid:15) Σ( I ) (cid:47) (cid:47) H k +1 ,λ ( I ) δ (cid:47) (cid:47) H k,µ ( I ) id (cid:15) (cid:15) H k,λ ( I ) F λ,µ ( I ) (cid:47) (cid:47) H k,µ ( I )Diagram 1where [ ⊕ π ∈ I Σ( π )] and [ T s ] are induced isomorphisms of the chain maps ⊕ π ∈ I Σ( π ) and T s respectively, andΦ’s are isomorphisms which came from the definition of connection matrix. See Lemma 5.1 for a precisedefinition of ∆ λ and Φ λ .Thus we have that the chain map T = T s ◦ ⊕ π ∈ I Σ( π ) covers the flow-defined Conley-index isomorphism F λ,µ , therefore T is an GTTM and T s is a GTM which covers F λ,µ ◦ Σ − = δ .Now we just need to prove that the latter diagram commutes. Indeed, H k,λ ( I ) id (cid:15) (cid:15) Σ( I ) (cid:47) (cid:47) H k +1 ,λ ( I ) δ (cid:47) (cid:47) H k,µ ( I ) id (cid:15) (cid:15) H k,λ ( I ) F λ,µ ( I ) (cid:47) (cid:47) H k,µ ( I )commutes by Theorem 5.2 and H k +1 ∆ Σ λ ( I ) Φ Σ λ ( I ) (cid:15) (cid:15) [ T s ] (cid:47) (cid:47) H k ∆ µ ( I ) Φ µ ( I ) (cid:15) (cid:15) H k +1 ,λ ( I ) δ (cid:47) (cid:47) H k,µ ( I )commutes by definition of ∆ g . The last diagram commutes by Lemma 5.1. Now one needs to check that the induced map [ T ] = (cid:2) T s ◦ (cid:76) π ∈ P Σ( π ) (cid:3) is a well defined map between thebraids H ∆ λ and H ∆ µ , indeed this follows since all maps from Diagram 1 are graded module braid maps. (cid:3) Note that Σ depends on the flow defined isomorphism F , therefore ∆ Σ λ and T s depends on F too, if oneis willing to remove that dependence (in order to obtain those maps in an easy way) the next corollary solvethis problem. Corollary 5.1.
The suspension isomorphism Σ does not depend on F when the flow at parameter λ isstructural stable. Moreover, T = T s ◦ ⊗ σ, where T is a transition matrix that covers the Conley flow-defined isomorphism F and T s is an MM-singulartransition matrix. Proof:
Let α be a path between λ to µ . If λ = µ , let α be a constant path, hence we have that Σ = × ◦ ⊗ σ ,therefore Σ does not depend on F and by Theorem 5.4 the result follows. Suppose that α is a non-constantpath, since the flow in parameter λ is structural stable there is a (cid:15) > α [0 , (cid:15) ] doesnot have bifurcations, thus F α (0)[0 ,(cid:15) ] = id , where α (0) is a constant path evaluated in α (0).By Λ being simply connected, we have that there exists a homotopy A : α (cid:39) ξ , where ξ = α (0)[0 , (cid:15) ] ∗ γ ∗ α [ (cid:15),
1] and γ is a path between α (0) and α ( (cid:15) ). Hence F α [0 ,(cid:15) ] = id , thus Σ does not depend on F , since wecan suspended Conley index by Σ locally. Therefore, by Theorem 5.4, the result follows. (cid:3) Theorem 6.4 in the next section shows us the importance when Σ does not depend on F , i. e., when onecan apply Corollary 5.1. 6. Topological Transition Matrix.
In this section, we develop more proprieties of generalized topological transition matrices from [FdRV]given that such matrices are transition matrix which covers flow-defined Conley-index isomorphisms. There-fore we can use the interrelationship between singular and topological transition matrix theory in orderto obtain refined properties, for instance, Theorem 6.3 is an improvement of Theorem 5 in [FdRV] by norequiring existence assumption. Moreover, Theorem 6.4 describe appropriately the importance of the unifytheory of transition matrix.Let M λ = { M λ ( π ) } π ∈ P and M µ = { M µ ( π ) } π ∈ P be Morse decompositions, related by continuation, forthe isolated invariant sets S λ ⊆ X λ and S µ ⊆ X µ , respectively. Definition 6.1. If T is a transition matrix that covers the Conley flow-defined isomorphism F , then werefer to T as a (generalized) topological transition matrix . In order to simplify one may omit the term in parenthesis “generalized” from Definition 6.1, neverthelessbe aware that this definition is an extension of topological transition matrix defined in [MM1], where theyassume that there are no connections at the initial and final parameters of a continuation.By the definition of transition matrix we obtain the following statement.
RANSITION MATRIX THEORY 21
Theorem 6.1.
Generalized topological transition matrix is a transition matrix which covers the flow-definedisomorphism F . (cid:3) Here is another way to characterize generalized topological transition matrices.
Proposition 6.1. T is a generalized topological transition matrix related to the connection matrices (∆ λ , Φ λ ) and (∆ µ , Φ µ ) , if and only if T : (cid:77) p ∈ P CH ∗ ( M λ ( p )) → (cid:77) p ∈ P CH ∗ ( M µ ( p )) is a zero degree map such that • { T ( I ) } I ∈ I ( < ) is a chain map from C ∆ λ to C ∆ µ ; • the following diagram commutes H ∆ λ ( I ) H ∆ λ ( IJ ) H ∆ λ ( J ) H ∆ λ ( I ) H λ ( I ) H λ ( IJ ) H λ ( J ) H λ ( I ) H ∆ µ ( I ) H ∆ µ ( IJ ) H ∆ µ ( J ) H ∆ µ ( I ) H µ ( I ) H µ ( IJ ) H µ ( J ) H µ ( I ) F λµ ( I ) ∆ λ ( J , I ) δ λ ( J , I ) ˆ T ( I ) Diagram 1 for all adjacent pairs ( I , J ) , where ˆ T ( · ) is the induced homology map of T ( · ) . Denote GTTM( < ) as the set of all generalized topological transition matrices with the partial order < .When there are no connections in the λ and µ parameters, then ∆ λ = 0 = ∆ µ . By Conley’s theory we havethat there is an isomorphism Φ λ : C ∗ ∆ λ ( P ) → H ∗ ,λ ( P ) for λ ∈ Λ (cid:48) , where C ∗ ∆ λ ( P ) = (cid:76) π ∈ P CH ( M λ ( π )) isthe chain complex with connection matrix ∆ λ .Therefore, we can carry out the continuation along the path ω in two ways: first by continuing S λ along thepath ω using the isomorphism F λ,µ ; secondly continuing (cid:83) p ∈ P M λ ( p ) along the path ω by using isomorphism E λ,µ = (cid:76) p ∈ P F λ,µ ( M ( p )). More precisely, we have the following diagram C ∆ λ ( P ) E λ,µ (cid:47) (cid:47) Φ λ (cid:15) (cid:15) C ∆ µ ( P ) Φ µ (cid:15) (cid:15) H λ ( P ) F λ,µ (cid:47) (cid:47) H µ ( P ) In general the diagram above is not commutative, in order to commute just define the following generalizedtopological transition matrix T λ,µ = Φ − µ ◦ F λ,µ ◦ Φ λ , then we obtain from Proposition 6.1 the followingdiagram commutes C ∆ λ ( P ) T λ,µ ( P ) (cid:47) (cid:47) Φ λ (cid:15) (cid:15) C ∆ µ ( P ) Φ µ (cid:15) (cid:15) H λ ( P ) F λ,µ ( P ) (cid:47) (cid:47) H µ ( P )In this particular case, such matrix T λ,µ is called a ( classical ) topological transition matrix .Applying Theorem 5.4 we have the following existence result. Theorem 6.2.
Let M λ = { M λ ( π ) } π ∈ ( P ,< λ ) and M µ = { M µ ( π ) } π ∈ ( P ,< µ ) be Morse decompositions, assumethat M λ and M µ are related by continuation with an admissible ordering < . Then there exists a generalizedtopological transition matrix T related to a ∆ λ ∈ CM ( < λ ) and a ∆ µ ∈ CM ( < µ ) . (cid:3) Now we can remove the existence hypotheses from item (v) of Theorem 5 in [FdRV] using Theorem 6.2.
Theorem 6.3.
Let M λ = { M λ ( π ) } π ∈ ( P ,< λ ) and M µ = { M µ ( π ) } π ∈ ( P ,< µ ) be Morse decompositions, ∆ λ and ∆ µ their respective connection matrices. Moreover, assume that M λ and M µ are related by continuationwith an admissible ordering < . Then the generalized topological transition matrix T satisfies the followingproperties: (i): T ◦ ∆ λ = ∆ µ ◦ T ; (ii): T λ,µ ( { p } ) = id and T is upper triangular with respect to < ; (iii): T is an isomorphism; (iv): T λ,λ = id , T λ,ν ( I ) = T µ,ν ◦ T λ,µ ( I ) and T µ,λ ( I ) = T − λ,µ ( I ) are generalized topological transitionmatrices, for all intervals I ∈ I and p ∈ P , in particular T = T ( P ) . (v): Let ω : [0 , → Λ be a path that continues M λ to M µ . Assume that T λ,µ ( p, q ) (cid:54) = 0 for allgeneralized topological transition matrices. Then there exists a finite sequence ≤ s ≤ s ≤ . . . ≤ s n ≤ and a sequence ( p i ) ⊆ P such that p = q, p n = p and the set of connecting orbits C (cid:0) M ω ( s i ) ( p i − ) , M ω ( s i ) ( p i ) (cid:1) is non-empty. Proof:
From Theorem 6.2 we have the existence of T λ,µ related to ¯∆ λ ∈ CM ( < λ ) and ¯∆ µ ∈ CM ( < µ ).However, ∆ α may be different from ¯∆ α , with α = { λ, µ } , since Theorem 6.2 does not specify which connectionmatrix T is related. Therefore, one needs to check in the proof of Theorem 5 in [FdRV] if all connectionmatrices at parameters ω (0) , ω ( s ξ ) , . . . , ω (1) agree.Without loss of generality, assume that we are in case where the connection matrices at parameters ω (0) = λ, ω ( s ξ ) = ν, ω (1) = µ must agree. Indeed, by Theorem 6.2 one obtains T λ,ν related to ∆ (cid:48) λ and ∆ (cid:48) ν and T ν,µ related to ∆ (cid:48)(cid:48) ν and ∆ (cid:48) µ . Although, ∆ (cid:48) ν may be different to ∆ (cid:48)(cid:48) ν , one can obtain the same matrix byusing Theorem 4.8 in [Fr2]. In other words, such theorem stats the existence of connection matrix by doingan induction process, since Theorem 6.2 use this result to obtain T then one can do the same induction RANSITION MATRIX THEORY 23 process starting from ∆ (cid:48) ν to obtain(5) ∆ = ∆ (cid:48) ν T (cid:48) ν,µ (cid:48)(cid:48) µ , where ∆ (cid:48)(cid:48) µ is a connection matrix at parameter µ and T (cid:48) ν,µ is a generalized topological transition matrixrelated to ∆ (cid:48) ν and ∆ (cid:48)(cid:48) µ .Note that, T ν,µ and ∆ (cid:48) µ may be different from T (cid:48) ν,µ and ∆ (cid:48)(cid:48) µ , respectively. But it is not a problem becauseone can obtain a T (cid:48) λ,µ = T λ,ν ◦ T (cid:48) ν,µ and by hypothesis T (cid:48) λ,µ ( p, q ) (cid:54) = 0. (cid:3) Remark 6.1.
Applying the same idea, in the previous proof, on Theorem 6.2 and on equation 3 and 4,one can freely choose a connection matrix (in their respective set of connection matrices) to be ∆ λ , ∆ − and ∆ α (0) respectively. However, ∆ µ , ∆ + and ∆ Σ α (1) are not free to choice, in fact they come from the inductiveprocess of Theorem 4.8 in [Fr2] . In other words, existence results coming from the singular transition matrixtheory give us liberty to choose a connection matrix at the parameter that was not suspended, yet we can notchoose freely a connection matrix at the parameter which was suspended. Using both topological and singular transition matrix, one is able to obtain a richer dynamical informationin Theorem 6.3 without assuming that T λ,µ ( p, q ) (cid:54) = 0 for all generalized topological transition matrices. Inother words, it is enough to assume T λ,µ ( p, q ) (cid:54) = 0 for at least one matrix in GTTM. Theorem 6.4.
Let M λ = { M λ ( π ) } π ∈ ( P ,< λ ) and M µ = { M µ ( π ) } π ∈ ( P ,< µ ) be Morse decompositions relatedby continuation with an admissible ordering < . Moreover, assume that the flow at parameter λ is structuralstable and a generalized topological transition matrix T has a nonzero entrance T λ,µ ( p, q ) (cid:54) = 0 . Then thereexists a finite sequence < s ≤ s ≤ . . . ≤ s n ≤ and a sequence ( p i ) ⊆ P such that p = q, p n = p andthe set of connecting orbits C (cid:0) M ω ( s i ) ( p i − ) , M ω ( s i ) ( p i ) (cid:1) is non-empty, where ω : [0 , → Λ is a path thatcontinues M λ to M µ . Proof:
By Corollary 5.1 a singular transition matrix is related to a generalized topological transition matrixby T λ,µ ( p, q ) = T s ◦ × ◦ ⊗ σ ( p, q )therefore T λ,µ ( p, q ) (cid:54) = 0 imply T s (cid:54) = 0. Since T s comes from a connection matrix ∆ g for a flow-defined orderit follows that a non zero entry in T s implies the result. (cid:3) Directional Transition Matrix
In this section, we prove that Direction Transition Matrix is a generalized transition matrix which coversan isomorphism defined via the flow-defined Conley index isomorphism F .Consider the fast-slow systems of the form (6) ˙ x = f ( x, y )˙ y = (cid:15)g ( x, y ) , where x ∈ R and y ∈ R . Observe that this fast-slow systems is more general than the other used todefine singular transition matrix. In contrast to equation 2 the slow variable y depends also on the fastvariable x , thus for (cid:15) > (cid:15) = 0 theparameterized system has an isolated invariant set S y continues over [0 ,
1] for each y ∈ [0 ,
1] and its Morsedecomposition M y = { M y ( π ) | π ∈ P } also continues over [0 , g ( M y ( π ) , y ) (cid:54) = 0 for all y ∈ (0 ,
1) and π ∈ P . Note thatdepending on g the slow dynamics introduced when (cid:15) > Definition 7.1.
A set B is a box if: (1) There exists an isolated neighborhood
B ⊆ R n × [0 , for the parameterized flow ψ B defined by ψ B : R × R n × [0 , → R n × [0 , t, x, y ) (cid:55)→ ( ψ y ( t, x ) , y ) , where ψ y is the flow of ˙ x = f ( x, y ) with fixed y . (2) Let S ( B = Inv ( B , ψ B ) . There is a Morse decomposition M ( S ( B )) = { M ( p, B ) | p = 1 , . . . , P B } , with the usual ordering on the integers as the admissible ordering. Let B y = B ∩ ( R n × { y } ) , S y ( B ) = Inv ( B y , ψ y ) and let { M y ( p, B | p = 1 , . . . , P B } be the corresponding Morse decomposition of S y ( B ) .Then S ( B ) = P B (cid:91) p =1 M ( p, B ) and S ( B ) = P B (cid:91) p =1 M ( p, B ) . (3) There are isolating neighborhood V ( p, B ) for M ( p, B ) such that V ( p, B ) ⊆ B and V ( p, B ) ∩ V ( q, B ) = ∅ for p (cid:54) = q with p, q = 1 , . . . , P B , and for every y ∈ [0 , V y ( p, B ⊆
Int ( B y ) . Furthermore, there are δ ( p, B ) ∈ {− , } , p = 1 , . . . , P B ) , such that δ ( p, B ) g ( x, y ) > , for all ( x, y ) ∈ V ( p, B ) . RANSITION MATRIX THEORY 25
From the last property, one can decompose the finite index set of the Morse decomposition as P = P + ∪ P − where P ± = { p ∈ P | ± δ ( p ) > } , and correspondingly, one can define M in ( p, B ) and M out ( p, B ) as follows: M in ( p, B ) = M ( p, B ) if p ∈ P + ,M ( p, B ) if p ∈ P − ; M out ( p, B ) = M ( p, B ) if p ∈ P + ,M ( p, B ) if p ∈ P − ;Notice that there are no connecting orbits among the Morse sets at y = 0 and at y = 1 and by theconstruction the sets S ( B ) and S ( B ) are related by continuation. A box with bidirectional slow dynamicscan naturally occur in various problems, for instance, in the FitzHugh-Nagumo equation. See [GKMOR] and[Ku] for more explanation. For this situation, either singular or topological transition matrix is not usefulsince they are both essentially unidirectional. Proposition 7.1.
Let V , V (cid:48) and W , W (cid:48) be mutually isomorphic finitely generated free Abelian groups, andlet A : V ⊗ W → V (cid:48) ⊗ W (cid:48) be an isomorphism. Suppose A is an upper triangular with the following block decomposition A = X Y Z where X : V → V (cid:48) and Z : W → W (cid:48) are isomorphisms, then the following maps are all upper triangularisomorphisms: A = X Y Z − Z − : V ⊗ W (cid:48) → V (cid:48) ⊗ W,A = X − − X − Y Z : V (cid:48) ⊗ W → V ⊗ W (cid:48) ,A = X − − X − Y Z − Z − : V (cid:48) ⊗ W (cid:48) → V ⊗ W. In order to have a map from M out to M in , one can repeatedly apply the Proposition 7.1 to the topologicaltransition matrix until obtain an isomorphism D : (cid:77) p ∈ P CH ∗ ( M out ( p )) → (cid:77) p ∈ P CH ∗ ( M in ( p )) . The matrix representation of this isomorphism is called by directional transition matrix , which has thefollowing property.
Theorem 7.1. [KMO]
Let D be the directional transition matrix for a box in the fast-slow system (6). Ifits ( p, q ) -entry D ( p, q ) is nonzero, then there exist a finite sequence { y i } k +1 i =1 in [0 , and a sequence { p i } in P satisfying ∂ ( p i +1 )( y i +1 − y i ) > for all i = 1 , . . . , k − and p = p > p > . . . > p k > p k +1 = q such that the corresponding parameterized flow at y = y i has a connecting orbit from M y i ( p i ) to M y i ( p i +1 ) . Simple examples show us that D depends on the choices of applying Proposition 7.1 repeatedly andtherefore led us to the question: which way should we obtain D from T ? Theorem 7.2 answers this questionby propounding that different D (obtained from same T ) cover different isomorphim, which is releated bythe choice made by applying Proposition 7.1 repeatedly.In this sense, the definition of directional transition matrix D is deeper than just a rearrangement Morsesets from a topological transition matrix. At first glance, one could think that is an artificial definition,however D is actually a transition matrix which covers an isomorphism.The next proposition seems to be a redundant way to define D , nevertheless this new way is really helpfulin the continuation context as one can see in Theorem 7.2 Proposition 7.2.
A directional transition matrix D : (cid:76) p ∈ P CH ∗ ( M out ( p )) → (cid:76) p ∈ P CH ∗ ( M in ( p )) can berepresented by an isomorphism ¯ D : (cid:76) p ∈ P CH ∗ ( M ( p )) → (cid:76) p ∈ P CH ∗ ( M ( p )) after doing a changing of base,moreover D and ¯ D are represented by the same matrix. Proof:
Recall M in ( p, B ) = M ( p, B ) if p ∈ P + ,M ( p, B ) if p ∈ P − , and M out ( p, B ) = M ( p, B ) if p ∈ P + ,M ( p, B ) if p ∈ P − , define R ,out = id ( p ) if M out ( p ) = M ( p ) ,F ( p ) if M out ( p ) = M ( p ) , and R in, = id ( p ) if M in ( p ) = M ( p ) ,F ( p ) if M in ( p ) = M ( p ) . Note that D is defined via a topological transition matrix, which is an isomorphism from a base B to thebase (cid:76) p ∈ P F ( p )( B ). Thus ¯ D = R in, ◦ D ◦ R ,out : (cid:76) p ∈ P CH ∗ ( M ( p )) → (cid:76) p ∈ P CH ∗ ( M ( p )) is just a RANSITION MATRIX THEORY 27 change of base since R − ,out = R in, . Observe that id covers (cid:76) p ∈ P F ( p ), in other words, (cid:76) p ∈ P F ( p ) doesnot give information about connecting orbits between Morse sets, therefore ¯ D and D are represented by thesame matrix. (cid:3) Same idea works, in Proposition 7.2, when one needs to change the map ¯ D : (cid:76) p ∈ P CH ∗ ( M ( p )) → (cid:76) p ∈ P CH ∗ ( M ( p )) for ¯ D (cid:48) : (cid:76) p ∈ P CH ∗ ( M ( p )) → (cid:76) p ∈ P CH ∗ ( M ( p )) by choosing another path orientationin the parameter space [0 , n is even.Applying Proposition 7.2 on Proposition 7.1 in the fast-slow systems setting, the next lemma gives uswhat kind of isomorphisms that the matrices in Proposition 7.1 covers. Lemma 7.1.
Let T = X Y Z be the topological transition matrix for the fast-slow systems in (6) when (cid:15) = 0 . For A = T in Proposition7.1 the matrices A , A and A are transition matrices which cover (cid:77) p ∈ I F ( p ) ⊕ F ( J ) ◦ F ( P ) ◦ (cid:77) p ∈ I F ( p ) ⊕ F ( J ) , F ( I ) ◦ (cid:77) p ∈ J F ( p ) ◦ F ( P ) ◦ F ( I ) ◦ (cid:77) p ∈ J F ( p ) , and F respectively, where I and J are interval such that P = IJ , X : (cid:76) p ∈ I CH ∗ ( M ( p )) → (cid:76) p ∈ I CH ∗ ( M ( p )) and Z : (cid:76) p ∈ J CH ∗ ( M ( p )) → (cid:76) p ∈ J CH ∗ ( M ( p )) . Proof:
Apply the same change of base done in Proposition 7.2 to the matrices A , A and A , furthermorenote that the connection matrices on parameter 0 and 1 are equal to zero. Thus A , A and A are chainmaps.Since A = T − so A is a transition matrix which covers F . For the others, firstly, let α : [0 , → [0 , α | [0 , / and α | [2 / , are subpath from the parameter 1 to 0 and α | [1 / , / is a subpathfrom 0 to 1. Observe that A = X Y Z − Z − = id Z − X Y Z id Z − ,A = X − − X − Y Z = X − − X − Y Z X − − X − Y Z X − − X − Y Z . Thus, for the path α , A and A cover (cid:77) p ∈ I F ( p ) ⊕ F ( J ) ◦ F ( P ) ◦ (cid:77) p ∈ I F ( p ) ⊕ F ( J ) , F ( I ) ◦ (cid:77) p ∈ J F ( p ) ◦ F ( P ) ◦ F ( I ) ◦ (cid:77) p ∈ J F ( p ) , respectively, since X , X − , Z , Z − , id ( I ) and id ( J ) are transition matrices which cover F ( I ), F ( I ), F ( J ), F ( J ), (cid:76) p ∈ I F ( p ) and (cid:76) p ∈ J F ( p ), respectively. (cid:3) The next theorem describes how direction transition matrix fits in transition matrix theory. Furthermoreit shows us the relation between the choices made by applying Proposition 7.1 on T to obtain D and whichisomorphism D must cover. Theorem 7.2.
Directional transition matrix is a generalized transition matrix which covers G n ◦ · · · ◦ G ◦ F ◦ G ◦ · · · ◦ G n , where G i is the isomorphism defined in Lemma 7.1 after applying it i times, and n is the number of timeneeded to apply Proposition 7.1 on topological transition matrix T in order to obtain direction transitionmatrix D . Proof:
Let T be the topological transition matrix for a box B for a fast-slow system 6. Suppose that oneneeds to apply n times Proposition 7.1 on T in order to obtain directional transition matrix D . Instead ofusing Proposition 7.1, one can use Lemma 7.1, but be aware that what was done by using Proposition 7.1must be done in the same way for Lemma 7.1. In i -th time that one applies Lemma 7.1, the new matrix willcovers G i ◦ · · · ◦ G ◦ F ◦ G ◦ · · · ◦ G i , thus the process ends for i = n . And to recover D , one just needs to apply Proposition 7.2. (cid:3) RANSITION MATRIX THEORY 29
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E-mail address : robert [email protected] Departamento de Matematica, Universidade Estadual de Campinas, 13083–859, Campinas, SP, Brazil
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