Transition matrix theory : Teoria da matriz de transição

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Ewerton Rocha Vieira

Transition Matrix Theory

Teoria da Matriz de Transição

CAMPINAS 2015

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Ficha catalográfica

Universidade Estadual de Campinas

Biblioteca do Instituto de Matemática, Estatística e Computação Científica Ana Regina Machado - CRB 8/5467

Vieira, Ewerton Rocha,

V673t VieTransition matrix theory / Ewerton Rocha Vieira. – Campinas, SP : [s.n.], 2015.

VieOrientador: Ketty Abaroa de Rezende.

VieTese (doutorado) – Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica.

Vie1. Conley, Índice de. 2. Matriz de conexão. 3. Sequências espectrais

(Matemática). 4. Morse, Teoria de. 5. Teoria dos sistemas dinâmicos. I. Rezende, Ketty Abaroa de,1959-. II. Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica. III. Título.

Informações para Biblioteca Digital

Título em outro idioma: Teoria da matriz de transição Palavras-chave em inglês:

Conley index Connection matrix

Spectral sequences (Mathematics) Morse theory

Theory of dynamical systems

Área de concentração: Matemática Titulação: Doutor em Matemática Banca examinadora:

Ketty Abaroa de Rezende [Orientador] Maria do Carmo Carbinatto

Oziride Manzoli Neto

Mariana Rodrigues da Silveira Marcio Fuzeto Gameiro

Data de defesa: 05-03-2015

Programa de Pós-Graduação: Matemática

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Abstract

In this thesis, we present a unification of the theory of algebraic, singular, topological and di-rectional transition matrices by introducing the (generalized) transition matrix which encompasses each of the previous four. Some transition matrix existence results are presented as well as the verification that each of the previous transition matrices are cases of the (generalized) transition matrix. Furthermore, we address how applications of the previous transition matrices to the Conley Index theory carry over to the (generalized) transition matrix. When this more general transition matrix satisfies the additional requirement that it covers flow-defined Conley-index isomorphisms, one proves algebraic and connection-existence properties. These general transition matrices with this covering property are referred to as generalized topological transition matrices and are used to consider connecting orbits of Morse-Smale flows without periodic orbits, as well as those in a continuation associated to a dynamical spectral sequence.

Keywords: Theory of dynamical systems, Morse theory, Conley index, connection matrix,

spectral sequences.

Resumo

Nessa tese, apresentamos uma unificação da teoria das matrizes de transição algébrica, sin-gular, topológica e direcional ao introduzir a matriz de transição (generalizada), a qual engloba todas as quatros citadas anteriormente. Alguns resultados de existência são apresentados bem como a verificação de que cada matriz de transição supracitada são casos particulares da matriz de transição (generalizada). Além disso, nós abordamos como as aplicações das quatros matrizes de transiçao, na teoria do índice de Conley, se traduzem para a matriz de transição (generalizada).

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Quando a matriz de transição (generalizada) satisfizer o requerimento adicional de cobrir o iso-morfismo do índice de Conley 𝐹 definido pelo fluxo, pode-se provar propriedades de existência e de conexão de órbitas. Essa matriz de transição com a propriedade de cobrir o isomorfismo 𝐹 é definida como matriz de transição topológica generalizada e a utilizamos para obter conexões de órbitas num fluxo Morse-Smale sem órbitas periódicas bem como para obter conexões de órbitas numa continuação associada à sequência espectral dinâmica.

Palavras-chave: Teoria dos sistemas Dinâmicos, teoria de Morse, índice de Conley, matrix de

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Acknowledgement

I would like to express my sincere gratitude to my advisor, Prof. Dr. Ketty Abaroa de Rezende, for the patient guidance, encouragement and advice she has provided throughout my time as her student. Her insights and comments were invaluable over the years. I appreciate the trust and confidence that she had placed in me.

I wish to thank Robert Franzosa who worked with me on two papers that resulted in this thesis and made mathematical research fun. It has been a real pleasure to work with him and I look forward to continue collaborating with him in the future.

I would like to thank Professor John Franks for many helpful discussions during one year at Northwestern University. Also, I want to thank him for his encouragement in helping me develop a forthcoming paper.

I would like to thank the following professors for helpful discussion during my PhD: Prof. Dr. Mariana Rodrigues da Silveira, Prof. Dr. Oziride Manzoli Neto, Prof. Dr. Marco Antônio Teixeira and Dra. Dahisy Valadão de Souza Lima. I also thank the members of the thesis committee who read this work and provided helpful comments.

This research would not have been possible without the financial assistance provided by FAPESP, and I am very thankful to this agency.

I also extend my appreciation to the Mathematics Department and administrative staff at State University of Campinas, Northwestern University and University of Maine.

I also want to thank my friends and colleagues at Unicamp, who have accompanied me through different stages of this journey. A special thanks goes to Cleber Colle, Dahisy Lima, Gus-tavo Grings, Igor Lima, Juliana Larrosa, Kamila Andrade, Mariana Villapouca, Matheus Brito, Matheus Bernardini, Paulo Henrique da Costa, Rafaela Prado, who made my time in the PhD program more fun and interesting, and for helping me keep on track and never giving up my goals. Finally and foremost, I owe special gratitude to my family for their continuous and uncondi-tional support during my thesis and throughout my life in general. Special thanks to Clotildes Remotto, Cristina Rocha, Suely Costa and Waldeci Vieira.

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Chapter 1 Introduction

A challenging question in the study of dynamical systems is that of the existence of global bifurcations. The difficulty in detecting such bifurcation orbits is the fact that one must analyze the dynamical system globally. Topological techniques for global analysis are, therefore, a perfect fit for such an investigation. In particular, the Conley index theory has been a valuable topological technique for detecting global bifurcations in dynamical systems [C], [CF], [Fr1], [Fr2], [Fr3] and [MM1]. This index is a standard tool in the analysis of invariant sets in dynamical systems, and its significance owes partly to the fact that it is invariant under local perturbation of a flow (the continuation property).

Typically, one does not investigate a single invariant set in a dynamical system but rather works with decompositions of a larger invariant set into invariant subsets and connecting orbits between them. The Morse decomposition is the standard such decomposition in the Conley index theory. Within the index theory there are matrices of maps defined between the Conley indices of invariant sets in a Morse decomposition, and these matrices (the connection matrices) provide information about connections that exist between sets in the decomposition. The connection matrices also have local invariance properties under continuation. Nevertheless, under global continuation sets of connection matrices can undergo change which usually means that the dynamical system has undergone a global bifurcation.

Connection matrices have been extensively studied and can be computed by numerical tech-niques [BM], [BR] and [E]. Their continuation properties have proven useful in detecting global bifurcations. In particular, the continuation theorem [Fr3] states that the connection matrices of an admissible ordering are invariant under local continuation. Yet, under global continuation, sets of connection matrices can undergo change. For instance, if there is a continuation between

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parameters with unique but different connection matrices, then within the continuation there must be a parameter value with nonunique connection matrices. At such a parameter value the system typically has a global bifurcation.

In other words, Morse decompositions and connection matrices provide a supporting structure within which global bifurcations can be detected, particularly via changes in the associated alge-braic structures. These differences that occur in connection matrices under continuation, which can naturally be identified algebraically, was the main motivation for the introduction of transition matrices as a combinatorial mechanism to keep track of these changes. These transition matrices have since appeared in the literature under several guises: singular [R], topological [MM1], and algebraic [FM]. These three types of matrices are defined differently (particularly under contrast-ing conditions) and have distinct properties. On the other hand, due to underlycontrast-ing similarities in the definitions and their corresponding properties, a unified theory for transition matrices has long been called for.

An initial approach to identify bifurcations via the interplay between local invariance and global change in connection matrices was due to Reineck in [R](1988). By introducing an artificial slow flow on the parameter space in a continuous family of dynamical systems, he obtained a map between the Conley indices of Morse decomposition invariant sets at the initial parameter value and those at the final parameter value in a continuation. This map is known as a singular transition matrix, and it has the feature that a nonzero entry can 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) used the continuation property of the Conley index, without introducing an artificial slow flow, to define matrices of maps between the Conley indices of Morse decomposition invariant sets at the initial and final parameter values in a continuation. Their maps, known as topological transition matrices, are naturally-defined maps on 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 was necessary to assume that connection matrices were trivial at the end parameter values. Nonetheless, as with the singular transition matrices, they were able to demonstrate that non-trivial topological transition matrix entries identify potential bifurcations that exist in the overall continuation. Furthermore, in [MM2] they established an equivalence between the singular transition matrices and the topological transition matrices in 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,

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not requiring the assumption that the connection matrices are trivial at the end parameter values. In their case, the general topological transition matrix 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. Given that connection matrices for a Morse decomposition are not unique, they raised the question of whether the nouniqueness could be understood via similarity transformations between connection matrices. Such transformations are algebraically defined, and - besides being associated with nonuniqueness of connection matrices for a particular Morse decomposition - can be exploited to track changes in connection matrices under flow continuation. They developed an existence result for algebraic transition matrices under the assumption of a “stackable” underlying partial order, and they demonstrated how algebraic transition matrices could 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 trans-formation that is similar in nature to both the singular transition matrix and the topological transition matrix. As in the singular transition matrix case, a slow flow is added to the parameter space, but it is more general than the specific flow used in defining the singular transition matrix. The advantage to the more-general approach is that it allows us to detect broader families of bifurcation orbits under continuation than those that are detected by the singular and topological transition matrices. The directional transition matrix is a transformation between indices of Morse decomposition sets at each end of the continuation, but not simply 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 the indices 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 of the above cases, the authors demonstrate how non-trivial directional transition entries identify bifurcations that occur under continuation.

While these four types of transition matrices are each defined differently and in different set-tings, they have in common that each is a Conley-index based algebraic transformation that tracks changes in index information under continuation and thereby identifies global bifurcations that could occur during the continuation. It is natural to expect that the theories could be unified in an overarching transition matrix theory, and that is the main purpose of this thesis. The basic idea for this general transition matrix is that it covers natural flow-defined index isomorphisms

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that arise under a continuation.

The spirit of this thesis is to unify the theory of four types of transition matrices which is defined differently and in distinct settings. Such general theory allow us to make use of the interrelationship between them to obtain new dynamical information.

The work reported here is a transcription∖compilation of the [FdRV, FV] in a different organized way as follows:

• Chapters 3, 4, 5, 7 and Section 6.4 are from the paper [FV];

• Section 2.3, Sections 6.1, 6.2 and 6.3 are from the paper [FdRV].

This thesis is organized in five chapters as follows. Chapter 2 is dedicated to the necessary back-ground material: Conley index theory, connection matrix theory, continuation, spectral sequences and Morse chain complex.

In Chapter 3, we introduce our general definition of the transition matrix, which is purely algebraic, and we address some of its properties as well as existence results. Such an algebraic foundation is needed to demonstrate how previous results using specific versions of the transition matrix carry over to our general setting.

In Chapter 4, we show that the algebraic matrix theory developed in [FM] is in fact a particular case of transition matrix as we have defined it. However, in general, algebraic transition matrix does not cover the flow-defined continuation isomorphism 𝐹 , therefore such matrix has less dynamical features compared with the others transitions matrices.

In Chapter 5, the singular transition matrix is defined via a particular fast-slow system. As one might notice, in practice, it is difficult to obtain singular transition matrices since it is only possible to compute it via the dynamics of the slow system. However, by showing that those matrices are transition matrices that cover an isomorphism, we actually use the singular transition matrix theory to assist in the development of other transition matrices.

In Chapter 6, we focus on a newly defined and more general transition matrix, which has the additional property that it covers flow-defined Conley-index isomorphisms. We refer to these matrices as generalized topological transition matrices and prove several properties they possess. In contrast to the classical case, we do not require that there are no connections at the initial and final parameters of a continuation. As a consequence we can apply this theory to a much more broader setting of dynamical systems. We also establish properties of the generalized topological transition matrices - including connecting orbit existence results - corresponding to those of the

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classical topological transition matrix. As an application, we employ this new theory to Morse-Smale flows without periodic orbits. In this setting one demonstrates uniqueness and provides a simple way to compute the generalized topological transition matrix. Also, we see how the generalized topological transition matrices can be obtained from a continuation associated to a dynamical spectral sequence.

In Chapter 7, the directional transition matrix, which is defined for more general fast-slow systems, is shown to be a particular case of the (generalized) transition matrix. Moreover, the unclear choices made in [KMO] to obtain the directional transition matrix, is clarified in our work by proving its relation to an isomorphism that the direction transition matrix covers.

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Chapter 2 Background

In this chapter, we present some background on algebraic topology tools and topological dy-namical systems which will be needed throughout this thesis.

An introduction of Conley index theory is presented in Section 2.1. In Section 2.2, we give a brief introduction to partial orders, graded module braids, Morse decompositions and connection matrices. Also in Section 2.3, we present some results on continuation associated to the Conley index. In Section 2.4, we present a short introduction to spectral sequences and to the Spectral Sequence Sweeping Algorithm. This algorithm recovers a spectral sequence associated to a chain complex. Additionally, we introduce in Section 2.5 the Morse chain complex needed in this thesis. The references for this chapter are [CdRS, Fr1, FdRV, MdRS, MM1, MM2, S, Sp].

2.1 Conley Index Theory

The Conley index theory has several applications in the study of the dynamics of a system, including the existence of periodic orbits in Hamiltonian systems, proof of chaotic behaviour in dynamical systems and bifurcation theory.

In classical Morse theory the gradient vector field of a Morse function on a compact manifold gives rise to a flow with a finite number of hyperbolic fixed points. Conley extended the Morse index by introducing a topological index for any isolated invariant set. Analogous to the decomposition of critical points in the Morse theory, Conley decomposed a flow in isolated invariant sets forming a Morse decomposition.

It is well known that Morse theory has the feature to relate local information (the Morse indices of the hyperbolic critical points) to global information (the homology of the compact manifold). This relation appears in the form of the Morse inequalities. In the Conley theory, the

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Morse inequalities are replaced by connection matrix. In this sense, the connection matrix is a central concept in this theory and enables one to investigate and prove the existence of heteroclinic connections between isolated invariant sets.

Now, we present a brief introduction to homotopy and homology Conley indices, Morse de-compositions, homology index braids and connection matrices. References for this section are [CdRS, C, Fr1, Fr2, Fr3, FV, S].

2.1.1 Conley Index of an Isolated Invariant Set

Let 𝑋 be a Hausdorff topological space and 𝜑𝑡 a continuous flow on 𝑋, i.e., a continuous map

𝜑 : 𝑋 × R → 𝑋, (𝑥, 𝑡) ↦→ 𝜑𝑡(𝑥), which satisfies 𝜑0 = id and 𝜑𝑠∘ 𝜑𝑡 = 𝜑𝑠+𝑡. We call a subset 𝑆 ⊂ 𝑋

aninvariant set under 𝜑 if 𝜑𝑡(𝑆) = 𝑆 for all 𝑡 ∈ R. Given a subset 𝑁 ⊂ 𝑋, let

Inv𝜑(𝑁 ) = {𝑥 ∈ 𝑁 | 𝜑𝑡(𝑥) ∈ 𝑁, ∀𝑡 ∈ R},

that is, Inv𝜑(𝑁 ) is the maximal invariant subset in 𝑁 . We say that a subset 𝑆 ⊂ 𝑋 is an

isolated invariant set with respect to the flow 𝜑𝑡 if there exists a compact set 𝑁 ⊂ 𝑋 such that

𝑆 = Inv𝜑(𝑁 ) ⊂ int(𝑁 ). In this case, 𝑁 is said to be an isolating neighbourhood for 𝑆. A

particular case of an isolating neighbourhood 𝑁 is anisolating block, where the exit set of the flow

𝑁− = {𝑥 ∈ 𝑁 | 𝜑[0,𝑡)(𝑥) * 𝑁, ∀𝑡 > 0} is closed.

Given an isolated invariant set 𝑆, an index pair for 𝑆 in 𝑋 is a pair of compact sets (𝑁, 𝐿) such that 𝐿 ⊂ 𝑁 and

(1) 𝑁 ∖𝐿 is an isolating neighborhood of 𝑆 in 𝑋, i.e., 𝑆 = Inv𝜑(𝑁 ∖𝐿) ⊂ int(𝑁 ∖𝐿);

(2) 𝐿 is positively invariant relative to 𝑁 , i.e., if 𝑥 ∈ 𝐿 and 𝜑(𝑥, [0, 𝑇 ]) ⊂ 𝑁 then 𝜑(𝑥, [0, 𝑇 ]) ⊂ 𝐿;

(3) and 𝐿 is the exit set of the flow in 𝑁 , i.e., if 𝑥 ∈ 𝑁 and 𝜑(𝑥, [0, ∞)) * 𝑁 then there exists

𝑇 > 0 such that 𝜑(𝑥, [0, 𝑇 ]) ⊂ 𝑁 and 𝜑(𝑥, 𝑇 ) ∈ 𝐿.

N L S N L S N L S

Not possible Not possible Possible

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Index pair always exists. In fact, given an isolated invariant set 𝑆, Conley proved in [C] that there exists an index pair for 𝑆. It is easy to see that the index pair is not uniquely determined, see Figure 2.1. But given two index pairs (𝑁1, 𝐿1) and (𝑁2, 𝐿2) for 𝑆, the pointed spaces 𝑁1/𝐿1

and 𝑁2/𝐿2, obtained by collapsing the exit sets 𝐿1 and 𝐿2, respectively, to a point, have the same

homotopy type.

We define the homotopy Conley index 𝐼(𝑆, 𝜑) of 𝑆 as the homotopy type of the pointed space

𝑁/𝐿 and we define the homology Conley index 𝐶𝐻(𝑆) of 𝑆 as the reduced homology of 𝑁/𝐿,

where (𝑁, 𝐿) is an index pair for 𝑆. Denote ℎ* the rank of the homology Conley index 𝐶𝐻(𝑆).

Consider the flow on R2 _{containing two saddles as in Figure 2.1. Let 𝑆 be the set consisting}

of the two saddles, then 𝑆 is an isolated invariant set, and the Conley index is well defined for

𝑆. Considering the index pair (𝑁, 𝐿) illustrated in Figure 2.1, one has that the homotopy Conley

index of 𝑆 is the wedge sum of two pointed one-spheres, i.e. 𝐼(𝑆) = ∑︀1_∨∑︀1_{. Observe that the}

Conley index does not depend on the index pair.

x y S =_{{x, y}} N L h(S0_{) = Σ}1_{∨ Σ}1 x y S ={x, y} N L h(S0_{) = Σ}1_{∨ Σ}1

Figure 2.1: Homotopy type of the space 𝑁/𝐿.

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case, it follows that 𝐻(𝑁, 𝐿) ∼= 𝐻(𝑁/𝐿). An index pair can always be modified to a regular index pair. We assume from now on that the index pairs are regular, since for some algebraic techniques it is more convenient to work with the pair (𝑁, 𝐿) than to work with the pointed space 𝑁/𝐿.

The Conley index generalizes the Morse index. The Conley index is the homotopy type of a pointed space and is well defined for all isolated invariant sets. In the case of non degenerate critical points, these two notions are related as follows: if 𝑥 is a singularity with Morse index 𝑘 then the Conley index of 𝑥 is the homotopy type of the 𝑘-sphere, i.e. 𝐼(𝑥) = ∑︀𝑘

. Figure 2.2 illustrates this relation for singularities in R3_.

Σ

1

Σ

2

Σ

0

Atractor

Repeller

Saddle

of index 1

Figure 2.2: Index pairs and Conley indices for singularities in R3_.

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2.2 Connection Matrix Theory

In this subsection, we present some definitions and results on partial orders, graded module braids, Morse decompositions and connection matrices. More details can be found in [Fr1] and [Fr2].

Throughout thesis thesis 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 (I1, ..., I𝑛) of mutually disjoint

nonempty intervals in < satisfying:

• ⋃︀𝑛

𝑖=1I𝑖 ∈ I(<);

• 𝜋 ∈ I𝑗, 𝜋′ ∈ I𝑘, 𝑗 < 𝑘 imply 𝜋′ ≮ 𝜋.

The collection of adjacent 𝑛-tuples of intervals in < is denoted I𝑛(<). An adjacent 2-tuple

of intervals is also called an adjacent pair of intervals. If <′ is either an extension of < or the restriction of < to an interval in <, then I𝑛(<′) ⊆ I𝑛(<). If (I, J) is an adjacent pair (2-tuple)

of intervals, then I ∪ J is denoted IJ. If (I1, . . . , I𝑛) ∈ I𝑛(<) and ⋃︀𝑛𝑖=1I𝑖 = I, then (I1, . . . , I𝑛) is

called a decomposition of I.

Definition 2.2.1. A graded module braid over < is a collection 𝒢 = 𝒢(<) of graded modules and

maps between the graded modules satisfying:

1. for each I ∈ I(<), there is a graded module 𝐺(I),

2. for each (I, J) ∈ I2(<), 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)→ 𝐺(J)𝑝 → 𝐺(I) → · · ·𝜕 is exact,

• if I and J are noncomparable, then 𝑝(JI, I)𝑖(I, IJ) = 𝑖𝑑|𝐺(I), • if (I, J, K) ∈ I3(<), then the following braid diagram commutes.

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G(I) G(IJK) G(K) G(J) G(IJ) G(JK) G(IJ) G(K) G(J) G(I) G(IJK) i i p p ∂ p ∂ p i ∂ i i i p ∂ ∂ ∂ i

Assume that 𝒢 and 𝒢′ are graded module braids over <.

Definition 2.2.2. An 𝑟-degree map 𝜃 : 𝒢 → 𝒢′ is a collection {𝜃(I)}I∈𝐼(<) of 𝑟-degree module homomorphisms 𝜃(I) : 𝐺(I) → 𝐺′_(I) _{such that the following diagram commutes for each (I, J) ∈}

I2(<) : · · · //𝐺𝑘(I) 𝑖 // 𝜃(I) 𝐺𝑘(IJ) 𝑝 _// 𝜃(IJ) 𝐺𝑘(J) 𝜕𝜆(J,I) // 𝜃(J) 𝐺𝑘−1(I) // 𝜃(I) · · ·

· · · //𝐺′_𝑘−𝑟(I) 𝑖 //𝐺′_𝑘−𝑟(IJ) 𝑝 //𝐺′_𝑘−𝑟(J) 𝜕𝜇(J,I)//𝐺′_{𝑘−𝑟−1}(I) //· · ·

If, futhermore, 𝜃(I) is an isomorphism for each I ∈ I(<), then 𝜃 is called an 𝑟-degree isomor-phism and 𝒢 and 𝒢′ _{are said to be 𝑟 isomorphic.}

Definition 2.2.3. A chain complex braid over < is a collection 𝐶 = 𝐶(<) of chain complexes and

chain maps satisfying:

1. for each I ∈ I(<), there is a chain complex 𝐶(I),

2. for each (I, J) ∈ I2(<), there are 0 degree maps 𝑖(I, IJ) : 𝐶(I) → 𝐶(IJ) and 𝑝(IJ, J) :

𝐶(IJ) → 𝐶(J) which satisfy: • 𝐶(I) 𝑖

→ 𝐶(IJ)→ 𝐶(J)𝑝 is weakly exact,

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• if (I, J, K) ∈ I3(<), then the following braid diagram commutes. C(I) C(IJ) C(J) C(IJK) C(JK) C(K) i i i p p p p i

Now assume that 𝐶 and 𝐶′ are chain complex braids over <.

Definition 2.2.4. An 𝑟-degree chain map 𝑇 : 𝐶 → 𝐶′ is a collection of 𝑟-degree maps 𝑇 (I) :

𝐶(I) → 𝐶′(I), I ∈ I(<), such that for each (I, J) ∈ I2(<) the following diagram commutes:

𝐶𝑘(I) 𝑖 _// 𝑇 (I) 𝐶𝑘(IJ) 𝑝 _// 𝑇 (IJ) 𝐶𝑘(J) 𝑇 (J) 𝐶_𝑘−𝑟′ (I) 𝑖 //𝐶_𝑘−𝑟′ (IJ) 𝑝 //𝐶_𝑘−𝑟′ (J)

Assume throughout the thesis that if the degree is not specified for a particular map, then it is 0-degree. The following proposition guarantees sufficient conditions for a map 𝑇 : 𝐶 → 𝐶′ to be a chain map.

Proposition 2.2.1. [Franzosa] Let 𝐶 = {𝐶(𝑝)}𝑝∈P and 𝐶′ = {𝐶′(𝑝)}𝑝∈P be collections of graded

modules, and Δ : ⨁︀ P𝐶(𝑝) → ⨁︀ P𝐶(𝑝) and Δ′ : ⨁︀ P𝐶′(𝑝) → ⨁︀ P𝐶′(𝑝) be <-upper triangular boundary maps. If 𝑇 :⨁︀ P𝐶(𝑝) → ⨁︀

P𝐶′(𝑝) is <-upper triangular and such that 𝑇 Δ = Δ′𝑇, then

{𝑇 (I)}I∈I(<) is a chain map from 𝒞Δ to 𝒞Δ′.

Let 𝜙 be a continuous flow on a locally compact Hausdorff space and let 𝑆 be a compact invariant set under 𝜙. A Morse decomposition of 𝑆 is a collection of mutually disjoint compact invariant subsets of 𝑆,

ℳ(𝑆) = {𝑀 (𝜋) | 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∖⋃︀

𝜋∈P𝑀 (𝜋) there exists 𝑝 < 𝑞 such that 𝛼(𝑥) ⊆ 𝑀 (𝑝)

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𝑀 , denoted <𝐹, and such that 𝑀 (𝜋) <𝐹 𝑀 (𝜋′) if and only if there exists a sequence of distinct

elements of 𝑃 : 𝜋 = 𝜋0, . . . , 𝜋𝑛 = 𝜋′, where 𝐶(𝑀 (𝜋𝑗), 𝑀 (𝜋𝑗−1)), the set of connecting orbit between

𝑀 (𝜋𝑗) and 𝑀 (𝜋𝑗−1), is nonempty for each 𝑗 = 1, . . . , 𝑛. Note that every admissible ordering of 𝑀

is an extension of <𝐹. Setting 𝑀 (I) = ⋃︁ 𝜋∈I 𝑀 (𝜋) ∪ ⋃︁ 𝜋,𝜋′_∈I 𝐶(𝑀 (𝜋′), 𝑀 (𝜋)),

the Conley index of 𝑀 (I), 𝐶𝐻*(𝑀 (I)), in short 𝐻*(I), is well defined, since 𝑀 (I) is an isolated

invariant set for all I ∈ I(<).

Given ℳ(𝑆), a Morse decomposition of 𝑆, the existence of an admissible ordering on ℳ(𝑆) implies that any recurrent dynamics in 𝑆 must be contained within the Morse sets, thus the dynamics off the Morse sets must be gradient-like. For this reason, Conley index theory refers to the dynamics within a Morse set as local dynamics and off the Morse sets as global dynamics. We briefly introduce the connection matrix theory, which addresses this latter aspect.

Definition 2.2.5. Given 𝒢, a graded module braid over <, and 𝐶 = {𝐶(𝑝)}𝑝∈P, a collection of

graded modules, let Δ :⨁︀

𝑝∈P𝐶(𝑝) →

⨁︀

𝑝∈P𝐶(𝑝) be a <-upper triangular boundary map. Then:

1. If ℋ, the graded module braid generated by Δ, is isomorphic to 𝒢, then Δ is called a

𝐶−connection matrix of 𝒢;

2. If, furthermore, 𝐶(𝑝) is isomorphic to 𝐺(𝑝) for each 𝑝 ∈ P, then Δ is called a connection

matrix of 𝒢.

To simplify notation, for I ∈ I(<) we denote⨁︀

𝜋∈I𝐶(𝜋) by 𝐶(I), and the corresponding

homol-ogy module in ℋΔ by 𝐻(I). In particular, the homolhomol-ogy 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 boundary map

Δ : ⨁︁

𝜋∈P

𝐶𝐻*(𝑀 (𝜋)) → ⨁︁

𝜋∈P

𝐶𝐻*−1(𝑀 (𝜋))

satisfying Definition 2.2.5 for 𝒞Δ = {𝐶𝐻*(𝑀 (𝜋))}𝜋∈P is called theconnection matrix for a Morse

decomposition. Moreover, let 𝒞ℳ(<) denote the set of all connection matrices for a given

(<-ordered) Morse decomposition ℳ(𝑆).

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Theorem 2.2.1. Let 𝒞(<) be a chain complex braid, and 𝐶 = {𝐶Δ(𝜋)}𝜋∈P be such that for each

𝜋 ∈ P, 𝐶Δ(𝜋) is a free chain complex with homology 𝐻Δ(𝜋) isomorphic to 𝐻(𝜋), the homology of the chain complex 𝐶(𝜋) in 𝒞(<). Then there exists an uppper triangular boundary map,

Δ : ⋃︁

𝜋∈P

𝐶Δ(𝜋) → ⋃︁

𝜋∈P

𝐶Δ(𝜋),

and a chain map, Ψ : 𝒞Δ(<) → 𝒞(<), from the chain complex braid defined by Δ to 𝒞(<), such that 𝜃 : ℋΔ(<) → ℋ𝒞(<), the map induced by Ψ, is an isomorphism.

Moreover, if <1 and <2 are admissible orderings of 𝒟(𝑆) such that <2 is an extension of <1,

then 𝒞ℳ(<1) ⊂ 𝒞ℳ(<2). In particularly, 𝒞ℳ(<𝐹) ⊂ 𝒞ℳ(<), for all admissible ordering < of

𝒟(𝑆).

The set 𝒞ℳ(<) provides some dynamical information about the structure of an invariant set

𝑆. A well known fact is that if Δ ∈ 𝒞ℳ(<𝐹), 𝜋 and 𝜋′ are adjacent in the flow ordering and

Δ(𝜋′, 𝜋) ̸= 0 then 𝐶(𝑀𝜋′, 𝑀_𝜋) ̸= ∅.

Note that algebraic properties of Δ put restrictions on the maps 𝜕(𝜋, 𝜋′). Δ can be used to prove the existence of connecting orbits between Morse sets. Furthermore, this theory can also be applied to the study of parameterized families of flows, according to the following two approaches: first by studying the stability of connection matrices under perturbations, whenever some stable connecting orbits are identified; and secondly by studying the changes in connection matrices under perturbation, whenever bifurcations are detected, see [Fr3] and [FdRS].

Example 1. Consider the flow illustrated in Figure 2.3, where the isolated invariant set 𝑆 consists

of singularities 𝑎, 𝑏 and 𝑐 and the connections between them. Let 𝑀1 = {𝑎}, 𝑀2 = {𝑏} and

𝑀3 = {𝑐}. 𝒟(𝑆) = {𝑀1, 𝑀2, 𝑀3} is a <-ordered Morse decomposition of 𝑆, where (𝑃, <) is the ordered set 𝑃 = {1, 2, 3} with 1 < 2 < 3. Note that the admissible ordering coincides with the flow order.

Now, we present a standard example that can be found in [Fr1, Fr2, L, Si, V]

The homotopy Conley index of the Morse sets 𝑀𝐼, with 𝐼 ∈ ℐ(<) are 𝐼(𝑀1) = Σ0, 𝐼(𝑀2) = 𝐼(𝑀3) = 𝐼(𝑀123) = Σ1, 𝐼(𝑀12) = ¯0, 𝐼(𝑀23) = Σ1 ∨ Σ1. Consider the module 𝐺 = Z2. Let

Δ : 𝐶Δ(𝑃 ) → 𝐶Δ(𝑃 ) be a strictly upper triangular boundary map

Δ = ⎛ ⎜ ⎜ ⎜ ⎝ 0 Δ(2, 1) Δ(3, 1) 0 0 Δ(3, 2) 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎠ ,

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S a b c M1 M2 M3

Figure 2.3: 𝑆 is an isolated invariant set.

where the homomorphisms Δ(2, 1) : 𝐻(𝑀2) → 𝐻(𝑀1), Δ(3, 1) : 𝐻(𝑀3) → 𝐻(𝑀1) and Δ(3, 2) : 𝐻(𝑀3) → 𝐻(𝑀2) are of degree −1. Of course Δ(3, 2) is the null map, since 𝐻(𝑀3) and 𝐻(𝑀2) are non zero only in dimension 1.

Since the Conley indices are computed in this example over Z2, then Δ(2, 1) is an isomorphism or 2 the null map. Analogously for Δ(3, 1).

In order for Δ to be a connection matrix for 𝒟(𝑆), the graded module braid ℋΔ generated by

Δ must be isomorphic to the homology index braid. From this fact, one can obtain information

about the maps Δ(2, 1) and Δ(3, 1). In this way, the homology of the complexes (𝐶Δ(𝐼), Δ(𝐼)), where 𝐼 ∈ ℐ and Δ(𝐼) is the restriction of Δ to the interval 𝐼, are:

𝐻𝑛Δ(𝑖) = ker Δ𝑛(𝑖) im Δ𝑛+1(𝑖) = 𝐻𝑛(𝑀𝑖) 0 ∼ = 𝐻𝑛(𝑀𝑖), for 𝑖 = 1, 2, 3, 𝐻𝑛Δ(12) = ker Δ𝑛(2, 1) im Δ𝑛+1(2, 1) = 𝐻𝑛(𝑀1) ⊕ ker Δ𝑛(2, 1) im Δ𝑛+1(2, 1) ⊕ 0 , (2.2.1) 𝐻𝑛Δ(23) = ker Δ𝑛(3, 2) im Δ𝑛+1(3, 2) = 𝐻𝑛(𝑀2) ⊕ 𝐻𝑛(𝑀3) 0 ∼ = 𝐻𝑛(𝑀2) ⊕ 𝐻𝑛(𝑀3).

Observe that the homology of (𝐶Δ(𝐼), Δ(𝐼)) is isomorphic to the homology Conley index of 𝑀𝐼,

for all interval 𝐼, except when 𝐼 = {1, 3}. Hence, one needs to guarantees that 𝐻Δ(12) ∼= 𝐻(𝑀12). Remember that Δ(2, 1) can be a null map or an isomorphism; since (2.2.1) must hold, then if

Δ(2, 1) = 0 one has 𝐻Δ(12) ∼= 𝐻(𝑀1) ⊕ 𝐻(𝑀2) 𝐻(𝑀12). But, if Δ(2, 1) is an isomorphism, then 𝐻Δ(12) ∼= 𝐻(𝑀12).

Note that there are no restrictions on the map Δ(3, 1) in order for Δ to be a connection matrix for the Morse decomposition 𝒟(𝑆). Therefore, the connections matrices for this example are the maps

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Δ = ⎛ ⎜ ⎜ ⎜ ⎝ 0 ≈ ≈ 0 0 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎠ and Δ = ⎛ ⎜ ⎜ ⎜ ⎝ 0 ≈ 0 0 0 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎠ .

2.3 Continuation

One of the key features in Conley theory is its invariance under continuation. Since the connec-tion matrices for Morse decomposiconnec-tions are algebraically derived from the homology Conley index braid, this seems to indicate that connecting orbits that persist over open sets in parameter space are identified by connection 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 Π𝑋 : 𝑋 × Λ → 𝑋 and ΠΛ : 𝑋 × Λ → Λ be the canonical projection

maps. See [S] and [Fr3].

Definition 2.3.1. A parametrization of a local flow 𝑋 ⊆ Γ is a homeomorphism 𝜑 : 𝑍 × Λ → 𝑋

such that for each 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 𝜑𝜆 and its image by 𝑋𝜆.

Lemma 2.3.1. [Salamon] For any compact set 𝑁 ⊆ 𝑋 the set Λ(𝑁) = {𝜆 ∈ Λ | 𝑁 × 𝜆 is an

isolating neighborhood in 𝑋 × 𝜆} is open in Λ.

Definition 2.3.2. 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 𝜚𝑁 : Λ(𝑁 ) →S and 𝜚𝑁(𝜆) = 𝐼𝑛𝑣(𝑁 × 𝜆). Then

consider the topology on the spaceS generated by the sets {𝜚𝑁(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 ΠΛ∘ 𝛾 = 𝑖𝑑|Λ.

We are interested in the situation where the homology index braids of admissible orderings of Morse decompositions at parameters 𝜆 and 𝜇 are isomorphic. That is, it is not enough that a Morse decomposition continues over Λ, it must also continue with a partial order, more specifically:

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Definition 2.3.3. Let ℳ(𝑆) = {𝑀(𝜋) | 𝜋 ∈ (P, <)} be an ordered Morse decomposition of the

isolated invariant set 𝑆 ⊆ 𝑋 × Λ. Let 𝑀𝜆 = {𝑀𝜇(𝜋)}𝜋∈(P,<𝜆), 𝑀𝜇 = {𝑀𝜇(𝜋)}𝜋∈(P,<𝜇), 𝑆𝜆 and 𝑆𝜇 be the sets obtained by intersection of ℳ(𝑆) and 𝑆 by the fibers 𝑋 ×{𝜆} and 𝑋 ×{𝜇}, respectively, where <𝜈 is the order 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 𝜍𝜋 : Λ →S

such that {𝜍𝜋(𝜈)| 𝜋 ∈ (P, <𝜈)} is a Morse decomposition for 𝜎(𝜈), ∀𝜈 ∈ Λ.

• If, furthermore, there exist a path 𝜔 : [0, 1] → Λ from 𝜆 to 𝜇; 𝜎(𝜆) = 𝑆𝜆; 𝜎(𝜇) = 𝑆𝜇;

𝜍𝜋(𝜆) = 𝑀𝜆(𝜋); 𝜍𝜋(𝜇) = 𝑀𝜇(𝜋); and if ℳ(𝑆) continues at least over 𝜔([0, 1]), then we say

that the admissible orderings <𝜆 and <𝜇 are related by continuation or continue from one to

the other. See Figure 2.4.

Λ

id

Mλ Sλ Mµ Sµ ς σ

Π

Λ

σ, ς

Figure 2.4: Sections from Definition 2.3.3

The following Lemma 2.3.2 is helpful to understand the previous Definition 2.3.3.

Lemma 2.3.2. [McCord, Mischaikov, Salamon]

• Let 𝛾 : Λ → S be a section, then 𝛾 is continuous if and only if 𝑆 = ⋃︁

𝜆∈Λ

𝛾(𝜆)

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• Let 𝑆 = ⋃︁ 𝜆∈Λ 𝜎(𝜆), 𝑀 (𝜋) = ⋃︁ 𝜆∈Λ 𝜍𝜋(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 its Morse decomposition if, and only if, ℳ(𝑆) with its order continues.

Suppose that 𝑆0 and 𝑆1 are invariant sets related by continuation in 𝑋𝜆0 and 𝑋𝜆1. Hence, there exists a map 𝜔 : [0, 1] → Λ such that 𝜔(0) = 𝜆0 and 𝜔(1) = 𝜆1 and an isolated invariant

set 𝑆 over 𝜔(𝐼) such that 𝑆𝜆𝑖 = 𝑆𝑖. The inclusion 𝑓𝑖 : 𝑋𝜆𝑖 → 𝑋 × 𝜔(𝐼) induces an isomorphism 𝐶𝐻*(𝑆𝑖)

𝑓𝑖*

−→ 𝐶𝐻*(𝑆), where 𝐶𝐻*(𝑆𝑖) and 𝐶𝐻*(𝑆) indicates the Conley homology indices of 𝑆𝑖

in 𝑋𝜆𝑖 and of 𝑆 in 𝑋 × 𝜔(𝐼), respectively. Thus, there is an isomorphism, called Conley

flow-defined isomorphism

𝐹𝜔 : 𝐶𝐻*(𝑆0)

𝑓_1*−1∘𝑓0*

−→ 𝐶𝐻*(𝑆1),

that depends on the endpoint-preserving homotopy class 𝜔. If 𝜋1(Λ) = 0 then 𝐹𝜔 is independent

of the path 𝜔 and one writes 𝐹𝜆1,𝜆2 instead of 𝐹𝜔. The Conley flow-defined isomorphism is well-behaved with respect to composition of paths: 𝐹𝜆,𝜆 = 𝑖𝑑, 𝐹𝜇,𝜈 ∘ 𝐹𝜆,𝜇 = 𝐹𝜆,𝜈 and 𝐹𝜆,𝜇 = 𝐹𝜇,𝜆−1. For

more details see [MM2] and [S]. To simplify notation, we denote 𝐶𝐻*(𝑀𝜈(I)) = 𝐻*,𝜈(I) or just 𝐶𝐻(𝑀𝜈(I)) = 𝐻𝜈(I), where I ∈ I(<𝜈) and 𝜈 ∈ {𝜆, 𝜇}.

Note that, by Lemma 2.3.2, Definition 2.3.3 is equivalent to the definitions of continuation with order presented in [S], [Fr3], [MM1] and [FM].

By Lemma 2.3.2, the minimal order <𝑚 for ℳ(𝑆) that continues over Λ is the flow defined

order for ℳ(𝑆). Note that if ℳ(𝑆) with order < continues then < extends <𝑚.

Proposition 2.3.1. If 𝑝 <𝑚 𝑞 then there exists 𝑠1, 𝑠2, . . . , 𝑠𝑛 ∈ [0, 1] and a sequence (𝑝𝑖) ⊆ P

such that 𝑝0 = 𝑞, 𝑝𝑛 = 𝑝 and the set of connecting orbits 𝐶

(︁

𝑀𝜔(𝑠𝑖)(𝑝𝑖−1), 𝑀𝜔(𝑠𝑖)(𝑝𝑖)

)︁

is non-empty, where 𝜔 : [0, 1] → Λ is a path between 𝜆 and 𝜇. We call these connections, unordered chain connections, in short, 𝑢𝑐𝑐.

Proof: Since a Morse set of ℳ(𝑆) is

𝑀 (𝜋) = ⋃︁

𝜆∈Λ

𝜍𝜋(𝜆)

then 𝑝 <𝑚 𝑞 implies that there is a sequence (𝑝𝑖) ⊆ P such that 𝑝0 = 𝑞, 𝑝𝑛 = 𝑝 and the

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𝑀 (𝑝𝑖−1) and 𝑀 (𝑝𝑖) occurs at some parameter in Λ. Therefore, we have the desired result whenever

ℳ(𝑆) continues over a path 𝜔 : [0, 1] → Λ.

Now we have the necessary framework to state the following results in [Fr3] and [FM], which we use subsequently.

We have the following continuation theorem for homology index braids of admissible orderings of Morse decompositions.

Theorem 2.3.1. [Franzosa] If the admissible orderings <𝜆 and <𝜇 are related by continuation,

then ℋ(<𝜆) and ℋ(<𝜇), the homology index braids of the admissible orderings, are isomorphic.

The next global continuation theorem for connection matrices of admissible orderings of Morse decompositions follows from the previous theorem.

Theorem 2.3.2. [Franzosa] If the admissible orderings <𝜆 and <𝜇 are related by continuation,

then 𝒞ℳ(<𝜆) = 𝒞ℳ(<𝜇).

The following proposition describes the relationship between connection matrices of Morse decompositions if the flow ordering of one Morse decompostion continues to an admissible ordering of another, since every admissible ordering is an extension of the flow ordering.

Proposition 2.3.2. [Franzosa] Let <1 and <2 be admissible orderings for ℳ(𝑆) and assume that

<1 is an extension of <2. Then

𝒞ℳ(<2) ⊆ 𝒞ℳ(<1).

The collection of connection matrices of a Morse decomposition is upper semicontinuous over the space of Morse decompositions and over the parameter space Λ.

Theorem 2.3.3. [Franzosa] There exists a neighborhood 𝑊 of 𝜆 in Λ such that if 𝜇 ∈ 𝑊 , then

𝑀𝜆 is related by continuation with order to a Morse decomposition 𝑀𝜇 of an isolated invariant set

in 𝑋𝜇, and for such 𝑀𝜇, 𝒞ℳ(𝑀𝜇) ⊆ 𝒞ℳ(𝑀𝜆).

The next proposition is not hard to verify and it can be found in [FM].

Proposition 2.3.3. [Franzosa, Mischaikow] Let 𝐶 = {𝐶(𝑝)}𝑝∈P and 𝐶′ = {𝐶′(𝑝)}𝑝∈P be

collec-tions of graded modules, and Δ : 𝐶(P) → 𝐶(P) and Δ′ _{: 𝐶}′_{(P) → 𝐶}′_(P) _{be <-upper triangular} boundary maps. If 𝑇 : 𝐶(P) → 𝐶′_(P) _{is <-upper triangular and such that 𝑇 Δ = Δ}′_𝑇_{, then}

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2.4 Spectral Sequence Sweeping Algorithm

In following subsections, we present the spectral sequence and Spectral Sequence Sweeping algorithm that will be needed in Section 6.3. The later was introduced in [CdRS] and in [MdRS] for the case of a chain complex over Z and F, respectively.

2.4.1 Spectral Sequence for a Chain Complex

Let 𝑅 be a principal ideal domain. A 𝑘-spectral sequence 𝐸 over 𝑅 is a sequence {𝐸𝑟, 𝜕𝑟}, for

𝑟 ≥ 𝑘, such that

1. 𝐸𝑟 _{is a bigraded module over 𝑅, i.e., an indexed collection of 𝑅-modules 𝐸}𝑟

𝑝,𝑞, for all 𝑝, 𝑞 ∈ Z;

2. 𝑑𝑟 _{is a differential of bidegree (−𝑟, 𝑟 − 1) on 𝐸}𝑟_{, i.e., an indexed collection of homomorphisms}

𝑑𝑟 _{: 𝐸}𝑟

𝑝,𝑞→ 𝐸𝑝−𝑟,𝑞+𝑟−1𝑟 , for all 𝑝, 𝑞 ∈ Z, and (𝑑𝑟)2 = 0;

3. for all 𝑟 ≥ 𝑘, there exists an isomorphism 𝐻(𝐸𝑟_{) ≈ 𝐸}𝑟+1_{, where}

𝐻𝑝,𝑞(𝐸𝑟) = Ker𝑑𝑟_{: 𝐸}𝑟 𝑝,𝑞 → 𝐸𝑝−𝑟,𝑞+𝑟−1𝑟 Im𝑑𝑟 _{: 𝐸}𝑟 𝑝+𝑟,𝑞−𝑟+1 → 𝐸𝑝,𝑞𝑟 . Note that if 𝐸𝑟

𝑝,𝑞 = 0 for a fixed pair of integers (𝑝, 𝑞), then 𝐸𝑝,𝑞𝑟+𝑎 = 0, for all integers 𝑎 ≥

0. Moreover, if we define 𝐸𝑟 𝑞 =

⨁︀

𝑠+𝑡=𝑞𝐸𝑠,𝑡𝑟 , then the differential 𝑑𝑟 induces a homomorphism

𝜕𝑟 : 𝐸𝑟

𝑞 → 𝐸𝑞−1𝑟 such that {𝐸𝑟, 𝜕𝑟} is a chain complex with 𝑞-th homology module equal to

⨁︀

𝑠+𝑡=𝑞𝐻(𝐸)𝑠,𝑡.

Let 𝑍_𝑝,𝑞𝑘 = 𝐾𝑒𝑟(𝑑𝑘_𝑝,𝑞 : 𝐸_𝑝,𝑞𝑘 → 𝐸𝑘

𝑝,𝑞−1) and 𝐵𝑝,𝑞𝑘 = 𝐼𝑚(𝑑𝑘𝑝,𝑞+1 : 𝐸𝑝,𝑞+1𝑘 → 𝐸𝑝,𝑞𝑘 ), then 𝐵𝑘 ⊆ 𝑍𝑘

and 𝐸𝑘+1 _{= 𝑍}𝑘_/𝐵𝑘_{. Now, define 𝑍(𝐸}𝑘+1₎

𝑝,𝑞 = 𝐾𝑒𝑟(𝑑𝑘+1𝑝,𝑞 : 𝐸𝑝,𝑞𝑘+1 → 𝐸 𝑘+1

𝑝−1,𝑞) and 𝐵(𝐸𝑘+1)𝑝,𝑞 =

𝐼𝑚(𝑑𝑘+1_𝑝+1,𝑞: 𝐸_𝑝+1,𝑞𝑘+1 → 𝐸𝑘+1

𝑝,𝑞 ). By the Noether Isomorphism Theorem, there exist bigraded modules

𝑍𝑘+1_{and 𝐵}𝑘+1_{of 𝑍}𝑘_{containing 𝐵}𝑘_{such that 𝑍(𝐸}𝑘+1₎

𝑝,𝑞 = 𝑍𝑝,𝑞𝑘+1/𝐵𝑝,𝑞𝑘 and 𝐵(𝐸𝑘+1)𝑝,𝑞= 𝐵𝑝,𝑞𝑘+1/𝐵𝑝,𝑞𝑘 ,

for all 𝑝, 𝑞 ∈ Z. Hence 𝐵𝑘 _{⊆ 𝐵}𝑘+1 _{⊆ 𝑍}𝑘+1 _{⊆ 𝑍}𝑘_{. By induction, one obtains submodules}

𝐵𝑘⊆ 𝐵𝑘+1 _{⊆ . . . ⊆ 𝐵}𝑟 _{⊆ . . . ⊆ 𝑍}𝑟 _{⊆ . . . ⊆ 𝑍}𝑘+1 _{⊆ 𝑍}𝑘_,

such that 𝐸𝑟+1 _{= 𝑍}𝑟_/𝐵𝑟_.

Consider the bigraded modules 𝑍∞ = ∩𝑟𝑍𝑟, 𝐵∞ = ∪𝑟𝐵𝑟 and 𝐸∞ = 𝑍∞/𝐵∞. We call the

latter module as the limit of the spectral sequence. We say that a spectral sequence 𝐸 = {𝐸𝑟, 𝜕𝑟} is convergent if given 𝑝, 𝑞 there is 𝑟(𝑝, 𝑞) ≥ 𝑘 such that for all 𝑟 ≥ 𝑟(𝑝, 𝑞), 𝑑𝑟

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is trivial. We say that a spectral sequence 𝐸 = {𝐸𝑟_{, 𝜕}𝑟_{} is} convergent in the strong sense if given

𝑝, 𝑞 ∈ Z there is 𝑟(𝑝, 𝑞) ≥ 𝑘 such that 𝐸𝑟 𝑝,𝑞≈ 𝐸

∞

𝑝,𝑞, for all 𝑟 ≥ 𝑟(𝑝, 𝑞).

Let (𝐶, 𝜕) be a chain complex. Anincreasing filtration 𝐹 on (𝐶, 𝜕) is a sequence of submodules

𝐹𝑝𝐶 of 𝐶 such that:

1. 𝐹𝑝𝐶 ⊂ 𝐹𝑝+1𝐶, for all integer 𝑝;

2. the filtration is compatible with the gradation of 𝐶, i.e. 𝐹𝑝𝐶 is a chain subcomplex of 𝐶

consisting of {𝐹𝑝𝐶𝑞}. · · · −→ Fp−1Cp+q−1 −→ FpCp+q−1 −→ Fp+1 Cp+q−1 −→ Fp+2 Cp+q−1 −→ · · · · · · −→ Fp−1Cp+q −→ FpCp+q −→ Fp+1 Cp+q −→ Fp+2 Cp+q −→ · · · · · · −→ Fp−1Cp+q+1 −→ FpCp+q+1 −→ Fp+1 Cp+q+1 −→ Fp+2 Cp+q+1 −→ · · · ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ... ... ... ... ... ... ... ... i ∂

We say that a filtration 𝐹 on 𝐶 is convergent if ∩𝑝𝐹𝑝𝐶 = 0 and ∪𝑝𝐹𝑝𝐶 = 𝐶. We say that it is

finite if there are 𝑝, 𝑝′

∈ Z such that 𝐹𝑝𝐶 = 0 and 𝐹𝑝′𝐶 = 𝐶. Also, it is said to be bounded below if for any 𝑞 there is 𝑝(𝑞) such that 𝐹𝑝(𝑞)𝐶𝑞 = 0.

Given a filtration on 𝐶, we define the associated bigraded module 𝐺(𝐶) as

𝐺(𝐶)𝑝,𝑞 =

𝐹𝑝𝐶𝑝+𝑞

𝐹𝑝−1𝐶𝑝+𝑞

.

A filtration 𝐹 on 𝐶 induces a filtration 𝐹 on 𝐻*(𝐶) defined by

𝐹𝑝𝐻*(𝐶) = ℑ [𝐻*(𝐹𝑝𝐶) → 𝐻*(𝐶)].

If the filtration 𝐹 on 𝐶 is convergent and bounded below then the same holds for the induced filtration on 𝐻*(𝐶).

The next theorem (see [Sp]) shows that we can associate a spectral sequence to a filtered chain complex whenever the filtration is convergent and bounded below.

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Theorem 2.4.1. Let 𝐹 be a convergent and bounded below filtration on a chain complex 𝐶. Then

there is a convergent spectral sequence with 𝐸_𝑝,𝑞0 = 𝐹𝑝𝐶𝑝+𝑞 𝐹𝑝−1𝐶𝑝+𝑞 = 𝐺(𝐶)𝑝,𝑞 and 𝐸𝑝,𝑞1 ≈ 𝐻𝑝+𝑞 (︃ 𝐹𝑝𝐶𝑝+𝑞 𝐹𝑝−1𝐶𝑝+𝑞 )︃

and 𝐸∞_{is isomorphic to the bigraded module 𝐺𝐻*}_(𝐶)_{associated to the induced filtration on 𝐻*}_(𝐶)_.

The proof of this theorem provides algebraic formulas for the modules 𝐸𝑟_{, which are}

𝐸_𝑝,𝑞𝑟 = 𝑍 𝑟 𝑝,𝑞 𝑍_{𝑝−1,𝑞+1}𝑟−1 + 𝜕𝑍_{𝑝+𝑟−1,𝑞−𝑟+2}𝑟−1 , where 𝑍_𝑝,𝑞𝑟 = {𝑐 ∈ 𝐹𝑝𝐶𝑝+𝑞| 𝜕𝑐 ∈ 𝐹𝑝−𝑟𝐶𝑝+𝑞−1}.

Observe that, 𝐸∞ does not determine 𝐻*(𝐶) completely, but

𝐸_𝑝,𝑞∞ ≈ 𝐺𝐻*(𝐶)𝑝,𝑞 =

𝐹𝑝𝐻𝑝+𝑞(𝐶)

𝐹𝑝−1𝐻𝑝+𝑞(𝐶)

.

However, whenever 𝐺𝐻*(𝐶)𝑝,𝑞 is free and the filtration is bounded, we have

⨁︁

𝑝+𝑞=𝑘

𝐺𝐻*(𝐶)𝑝,𝑞 ≈ 𝐻𝑝+𝑞(𝐶). (2.4.1)

See [D].

2.4.2 Spectral Sequence Sweeping Algorithm

In [CdRS], it was defined a sweeping algorithm which recovers the spectral sequence associated to a finite chain complex over Z with a special filtration. More specifically, consider a finite chain complex (𝐶, 𝜕) such that each module 𝐶𝑘 is finite generated. Denote the generators of the 𝐶𝑘

chain module by ℎ1

𝑘, · · · , ℎ 𝑐𝑘

𝑘 . We can reorder the set of the generators of 𝐶* as

{ℎ1 0, · · · , ℎ ℓ0 0 , ℎ ℓ0+1 1 , · · · , ℎ ℓ1 1 , · · · , ℎ ℓ𝑘−1+1 𝑘 , · · · , ℎ ℓ𝑘 𝑘 , · · · },

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where ℓ𝑘 = 𝑐0+ · · · + 𝑐𝑘. Let 𝐹 be a finest filtration on 𝐶 defined by 𝐹𝑝𝐶𝑘 = ⨁︁ ℎℓ 𝑘, ℓ≤𝑝+1 Z⟨ℎℓ𝑘⟩,

for 𝑝 ∈ N. The spectral sequence associated to (𝐶*, 𝜕*) with this finest filtration has a special property: the only 𝑞 for which 𝐸𝑟

𝑝,𝑞 is non-zero is 𝑞 = 𝑘 − 𝑝, where 𝑘 is the index of the chain in

𝐹𝑝𝐶 ∖ 𝐹𝑝−1𝐶. Hence, in this case, we omit reference to 𝑞. It is understood that 𝐸𝑝𝑟 is in fact 𝐸𝑝,𝑘−𝑝𝑟 .

The sweeping algorithm presented below, provides an alternative way to obtain such modules as well as the differentials 𝑑𝑟’s.

With this in mind, we view the boundary map 𝜕 of the chain complex 𝐶 as the matrix Δ:

∆k−1 ∆k ∆k+1 ∆k+2 Ck Ck−1 Ck+1 Ck+2 Cn C0 0 Ck Ck−1 Ck+1 Ck+2 Cn C0 0 0 0 0 0 0 0 0 0 0 0 ... ... · · · · ∆ =

where Δ𝑘 is the map 𝜕𝑘 and the order of the columns of Δ follows the order determined on

the generators of 𝐶*. From now on, the boundary operator 𝜕 and the matrix Δ will be used

interchangeably. Note that the numbering on the columns of Δ is shifted by one with respect to the subindex 𝑝 of the filtration 𝐹𝑝. In order to simplify notation, we use the index 𝑓𝑘 to denote the

first column of Δ associated to a 𝑘-chain. Hence 𝑓𝑘 = ℓ𝑘−1 + 1. Moreover, ℓ𝑘 denotes the latter

column associated to a 𝑘-chain. The term ℎ𝑙

𝑘 denotes an elementary 𝑘-chain of the module 𝐶𝑘 and this 𝑘-chain is associated to

the column 𝑙 of the matrix Δ. Moreover, 𝐹𝑙−1∖ 𝐹𝑙−2 = Z⟨ℎ𝑙𝑘⟩.

Now, we present the sweeping algorithm. For more details see Theorems 4.4 and 5.7 in [CdRS].

Spectral Sequence Sweeping Algorithm - SSSA

For a fixed diagonal 𝑟 parallel and to the right of the main diagonal, the method described below must be applied simultaneously for all 𝑘.

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Initial Step.

(1) Let 𝜉1 be the first diagonal of Δ that contains non-zero entries Δ𝑖,𝑗 in Δ𝑘, which will be

called index 𝑘 primary pivots. Define Δ𝜉1 _{to be Δ with the 𝑘- index} primary pivots marked on the 𝜉1-th diagonal.

(2) Consider the matrix Δ𝜉1_{. Let 𝜉}

2 be the first diagonal greater than 𝜉1 which contains non-zero

entries Δ𝜉1

𝑖,𝑗. The construction of Δ𝜉2 follows the procedure below. Given a non-zero entry

Δ𝜉1

𝑖,𝑗 on the 𝜉2-th diagonal of Δ𝜉1:

If Δ𝜉1

𝑠,𝑗 contains an index 𝑘 primary pivot for 𝑠 > 𝑖, then the numerical value of the given

entry remains the same, Δ𝜉2

𝑖,𝑗 = Δ 𝜉1

𝑖,𝑗, and the entry is left unmarked.

If Δ𝜉1

𝑠,𝑗 does not contain a primary pivot for 𝑠 > 𝑖:

then if Δ𝜉1

𝑖,𝑡 contains a primary pivot, for 𝑡 < 𝑗,

then define Δ𝜉2

𝑖,𝑗 = Δ 𝜉1

𝑖,𝑗 and mark the entry Δ 𝜉2

𝑖,𝑗 as a change-of-basis pivot.

Else, define Δ𝜉2

𝑖,𝑗 = Δ 𝜉1

𝑖,𝑗 and permanently mark Δ 𝜉2

𝑖,𝑗 as an index 𝑘 primary pivot.

Intermediate Step.

Suppose by induction that Δ𝜉 is defined for all 𝜉 ≤ 𝑟 with the primary and change-of-basis pivots marked on the diagonals smaller or equal to 𝜉. In what follows it will be shown how Δ𝑟+1is defined. Without loss of generality, one can assume that there is at least one change-of-basis pivot on the

𝑟-th diagonal of Δ𝑟_{. If it is not the case, define Δ}𝑟+1 _{= Δ}𝑟_{with primary pivots and change-of-basis}

pivots marked as in step (2) below.

(1) Change of basis. Let Δ𝑟

𝑖,𝑗 be a change-of-basis pivot in Δ𝑟𝑘. Perform the change of basis

on Δ𝑟 _{by adding a linear combination over Q of all the ℎ}𝑙_𝑘 columns of Δ𝑟 with ℓ < 𝑗 to a positive integer multiple 𝑢 ̸= 0 of column 𝑗 of Δ𝑟_{, in order to zero out the entry Δ}𝑟

𝑖,𝑗 without

introducing non-zero entries in Δ𝑟

𝑠,𝑗 for 𝑠 > 𝑖. Moreover, the resulting linear combination

should be of the form 𝛽𝑓𝑘_ℎ𝜅

𝑘 + · · · + 𝛽𝑗−1ℎ 𝑗−1

𝑘 + 𝛽𝑗ℎ 𝑗

𝑘, where 𝑓𝑘 is the first column of Δ𝑟

associated to a 𝑘-chain and 𝛽ℓ _{∈ Z, for all 𝑗 = 𝜅, · · · , 𝑗.}

The integer 𝑢 is called the leading coefficient of the change of basis. If more than one linear combination is possible, one must choose the one which minimizes 𝑢. One can define a matrix 𝑇𝑟 _{which performs all the change of basis on all of the 𝑟-th diagonal. Define}

Δ𝑟+1 _{= (𝑇}𝑟₎−1_Δ𝑟_𝑇𝑟 _{and mark the entries of the (𝑟 + 1)-th diagonal of Δ}𝑟+1 _{as follows.}

(2) Markup. Given a non-zero entry Δ𝑟+1_𝑖,𝑗 on the (𝑟 + 1)-th diagonal of Δ𝑟+1_𝑘 :

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If Δ𝑟+1_𝑠,𝑗 does not contain a primary pivot for 𝑠 > 𝑖:

then if Δ𝑟

𝑖,𝑡 contains a primary pivot, for 𝑡 < 𝑗,

then mark Δ𝑟_𝑖,𝑗 as a change-of-basis pivot.

Else permanently mark Δ𝑟_𝑖,𝑗 as a primary pivot.

Final Step.

Repeat the above procedure until all diagonals have been considered.

According to the algorithm, if Δ𝑟

𝑖,𝑗 is a change-of-basis pivot on the 𝑟-th diagonal of Δ𝑟𝑘, then

once the corresponding change of basis has been performed, one obtains a new 𝑘-chain associated to column 𝑗 of Δ𝑟+1_{, which will be denoted by 𝜎}𝑗,𝑟+1

𝑘 . Observe that 𝜎 𝑗,𝑟+1

𝑘 is a linear combination

over Q of columns ℓ of Δ𝑟 _{with 𝑓}

𝑘 ≤ ℓ ≤ 𝑗 such that Δ𝑟+1𝑖,𝑗 = 0, i.e., 𝜎 𝑗,𝑟+1 𝑘 is a linear combination over Q of 𝜎𝑓𝑘,𝑟 𝑘 , · · · , 𝜎 𝑗,𝑟 𝑘 . Also, 𝜎 𝑗,𝑟+1

𝑘 is a linear combination over Z of the columns 𝑓𝑘, · · · , 𝑗 of Δ𝑟,

i.e., of ℎ𝑓𝑘 𝑘 , · · · , ℎ 𝑗 𝑘. Hence, 𝜎_𝑘𝑗,𝑟+1 = 𝑢 𝑗 ∑︁ ℓ=𝑓𝑘 𝑐𝑗,𝑟_ℓ ℎℓ_𝑘 ⏟ ⏞ 𝜎_𝑘𝑗,𝑟 + 𝑞𝑗−1 𝑗−1 ∑︁ ℓ=𝑓𝑘 𝑐𝑗−1,𝑟_ℓ ℎℓ_𝑘 ⏟ ⏞ 𝜎𝑗−1,𝑟_𝑘 + · · · + 𝑞𝑓𝑘+1𝑐 𝑓𝑘+1,𝑟 𝑓𝑘 ℎ 𝑓𝑘 𝑘 + 𝑐 𝑓𝑘+1,𝑟 𝑓𝑘+1 ℎ 𝑓𝑘+1 𝑘 ⏟ ⏞ 𝜎_𝑘𝑓𝑘+1,𝑟 + 𝑞𝑓𝑘𝑐 𝑓𝑘,𝑟 𝑓𝑘 ℎ 𝑓𝑘 𝑘 ⏟ ⏞ 𝜎𝑓𝑘,𝑟_𝑘 (2.4.2) = cj,r+1_k ℎ𝑗_𝑘+ cj−1,r+1_k ℎ𝑗−1_𝑘 + · · · + cfk,r+1 k ℎ 𝑓𝑘 𝑘

where cℓ,r+1_k _{∈ Z, for ℓ = 𝑓}𝑘, · · · , 𝑗. If Δ𝑟 contains an index 𝑘 primary pivot in the entry Δ𝑟_𝑠,¯_ℓ with

𝑠 > 𝑖 and ¯ℓ < 𝑗, then 𝑞¯_ℓ = 0. Of course, the first column of any Δ𝑘 cannot undergo changes of

basis, since there is no column to its left associated to a 𝑘-chain.

The family of matrices {Δ𝑟} produced by the Spectral Sequence Sweeping Algorithm has several properties, which are proven in [CdRS, MdRS], such as:

(a) Δ𝑟 _{is a strictly upper triangular boundary map, for each 𝑟.}

(b) It is not possible to have more than one primary pivot in a fixed row or column.

(c) If the entry Δ𝑟

𝑗−𝑟,𝑗 is a primary pivot or a change-of-basis pivot, then Δ𝑟𝑠,𝑗 = 0 for all 𝑠 > 𝑗 −𝑟.

(d) If Δ𝑟

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(e) Let Δ𝐿 _{be the last matrix produced by the SSSA. Then, the primary pivots are non-null and}

each non-null entry is located above a unique primary pivot.

(f) If column 𝑗 of Δ𝐿 _{is non-null, then row 𝑗 is null. The matrix Δ}𝐿 _{is integral.}

Example 2. To illustrate the SSSA over Z, consider the graded group 𝐶* defined by 𝐶0 _{= Z⟨ℎ}1 0⟩, 𝐶2 = Z⟨ℎ22⟩, 𝐶3 = Z⟨ℎ33⟩ ⊕ Z⟨ℎ43⟩ ⊕ Z⟨ℎ53⟩, 𝐶4 = Z⟨ℎ64⟩ ⊕ Z⟨ℎ74⟩, 𝐶6 = Z⟨ℎ86⟩ and 𝐶𝑘 = 0, for

𝑘 ∈ N ∖ {0, 2, 3, 4, 6}. Also, consider the differential 𝜕𝑘 : 𝐶𝑘 → 𝐶𝑘−1, defined on the generators of

𝐶𝑘 by 𝜕3(ℎ33) = 3ℎ22, 𝜕3(ℎ43) = 2ℎ22, 𝜕3(ℎ53) = ℎ22, 𝜕4(ℎ64) = 2ℎ33− 4ℎ43+ 2ℎ53, 𝜕4(ℎ74) = 1ℎ33+ 2ℎ43− 7ℎ53, and the other 𝜕𝑘 are the null map. The pair (𝐶*, 𝜕*) is a finite chain complex.

Applying the SSSA to this complex, one obtains the sequence of matrices Δ𝑟 shown in Figures

2.5 to 2.10. In these figures, the primary pivots entries are indicated by means of a red background and darker edge, the change-of-basis pivots are indicated by blue background and dashed edges, null entries are left blank and the diagonal being swept is indicated with a gray line.

h68 h01 h47 h22 h46 h33 h35 h34 h34 h35 h33 h46 h22 h47 h01 h68 3 2 1 2 1 -4 2 2 -7

Figure 2.5: Initial matrix Δ.

Σ68,1=h68 Σ₄7,1₌_h 4 7 Σ₄6,1₌_h 4 6 Σ3 5,1 =h35 Σ₃4,1₌_h 3 4 Σ33,1=h33 Σ₂2,1₌_h 2 2 Σ₀1,1₌_h 0 1 Σ0 1,1 Σ2 2,1 Σ3 3,1 Σ3 4,1 Σ3 5,1 Σ4 6,1 Σ4 7,1 Σ6 8,1 3 2 1 2 1 -4 2 2 -7

Transition matrix theory : Teoria da matriz de transição

Ewerton Rocha Vieira