2.1 Attractor–repeller decompositions and connection matrices

Throughout this paper let $(P,<)$ be a partial ordered set with partial order $<$ , where P is a finite set of indices. An interval in $<$ is a subset $I \subseteq P$ such that if $p,q \in I$ and $p < r < q$ then $r\in I$ . The set of intervals in $<$ is denoted by $I(<)$ .

An adjacent n-tuple of intervals in $<$ is an ordered collection $(I_1,\ldots ,I_n)$ of mutually disjoint non-empty intervals in $<$ satisfying:

• • $\bigcup _{i=1}^n I_i\in I(<);$

• • $\pi \in I_j, \pi '\in I_k, j<k$ imply $\pi '\nless \pi .$

The collection of adjacent n-tuples of intervals in $<$ is denoted $I_n(<)$ . An adjacent 2-tuple of intervals is also called an adjacent pair of intervals. If $<'$ is an extension of $<$ , then $I_n(<')\subseteq I_n(<)$ , for each n. If $<_{I}$ is a restriction of $<$ to an interval $I\in I(<)$ , then $I_n(<_{I})\subseteq I_n(<)$ , for each n. If $(I,J)$ is an adjacent pair (2-tuple) of intervals, then $I\cup J$ is denoted by $IJ$ . If $(I_1,\ldots , I_n)\in I_n(<)$ and $\bigcup _{i=1}^n I_i=I$ , then $(I_1,\ldots , I_n)$ is called a decomposition of I.

Let $\varphi : \mathbb {R}\times X \rightarrow X$ be a continuous flow on a locally compact Hausdorff space X and let $S\subseteq X$ be an invariant set under $\varphi $ . We use the notation $x\cdot t:=\varphi (t,x)$ . For any set $Y\subseteq S$ , the $\omega $ -limit and $\alpha $ -limit sets are given by $\omega (Y)=\bigcap _{t>0}\overline {Y\cdot [t,\infty )}$ and $\alpha (Y)=\bigcap _{t>0}\overline {Y\cdot (-\infty ,-t]}$ , respectively. Both sets are closed, and if S is compact then they will be compact. An invariant set $A\subseteq S$ is an attractor in S if there exists a S-neighborhood U of A such that $\omega (U)=A$ . A repeller in S is an invariant set $R\subseteq S$ such that there exists a S-neighborhood U of R with $\alpha (U) = R$ . Whenever S is a compact set then A and R are also compact sets.

A ( $<$ -ordered) Morse decomposition of S is a collection $ \mathcal {M}(S)=\{M(\pi )\ |\ \pi \in P\} $ of mutually disjoint compact invariant subsets of S, indexed by a finite set P, such that if $x\in S\backslash \bigcup _{\pi \in P}M(\pi )$ then there exist $\pi <\pi '$ such that $\alpha (x)\subseteq M(\pi ')$ and $\omega (x)\subseteq M(\pi ).$ Each set $M(\pi )$ is called a Morse set. A partial order on P with this property induces a partial order on $\mathcal {M}(S)$ called an admissible ordering of the Morse decomposition.

The flow defines an admissible ordering on $\mathcal {M}(S)$ , called the flow ordering of $\mathcal {M}(S)$ , denoted $<_F$ , such that $M(\pi )<_F M(\pi ') $ if and only if there exists a sequence of distinct elements of $P: \pi = \pi _0,\ldots ,\pi _n=\pi '$ , where the set of connecting orbits between $M(\pi _j)$ and $M(\pi _{j-1})$ ,

$$ \begin{align*} &C(M(\pi_j),M(\pi_{j-1}))\\ &\quad=\{x \in S\backslash (M(\pi_j)\cup M(\pi_{j-1})) \mid \alpha(x)\subseteq M(\pi_j) \text{ and } \omega(x)\subseteq M(\pi_{j-1})\}, \end{align*} $$

is non-empty for each $j = 1, \ldots ,n$ . Note that every admissible ordering of M is an extension of $<_F$ .

Given a Morse decomposition $\mathcal {M}(S)$ of S, the existence of an admissible ordering on $\mathcal {M}(S)$ 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 Conley index of an isolated invariant set and the connection matrix theory, which addresses this latter aspect. Recall that $S\subseteq X$ is an isolated invariant set if there exists a compact set $N\subseteq X$ such that $S \subseteq \mathrm {int} (N)$ and

$$ \begin{align*}S=\mathrm{Inv}(N,\varphi)=\{x\in N \mid\ \varphi(\mathbb{R},x) \subseteq N\}. \end{align*} $$

In this case N is said to be an isolating neighborhood for S in X. Note that isolated invariant sets are compact sets. An index pair for an isolated invariant set S is a pair $(N_1, N_0)$ of compact sets $N_0 \subseteq N_1$ such that: (i) $S \subseteq \mathrm {int}(N_1\backslash N_0)$ and $\textrm {cl}(N_1\backslash N_0)$ is an isolating neighborhood for S; (ii) $N_0$ is positively invariant in $N_1$ , that is, given $x \in N_0$ and $t>0$ such that $x \cdot [0, t] \subseteq N_1$ , then $x\cdot [0,t]\subseteq N_0$ ; (iii) $N_0$ is an exit set for $N_1$ , that is, given $x \in N_1$ and $t_1> 0$ with $x\cdot t_1 \notin N_1$ , there exists $t_0\in [0, t_1] $ such that $x\cdot [0, t_0]\subseteq N_1$ and $x\cdot t_0\in N_0$ . The theorems of existence and equivalence of index pairs guarantee that given any isolating neighborhood $N \subseteq X$ of S and any neighborhood U of S, there exists an index pair $(N_1, N_0)$ for S in X such that $N_1$ and $N_0$ are positively invariant in N and $cl(N_1 \setminus N_0) \subseteq U$ . Moreover, the homotopy type of the pointed space $N_1/N_0$ is independent of the choice of the index pair and therefore it only depends on the behavior of the flow near the isolated invariant set S. For more details, see [Reference Conley4, Reference Salamon20].

The homology Conley index of S, $CH_\ast (S)$ , is the homology (computed with coefficients in a module over a principal ideal domain) of the pointed space $N_1/N_0$ , where $(N_1,N_0)$ is an index pair for S. Setting

$$ \begin{align*}M(I)=\bigcup_{\pi\in I}M(\pi)\cup\bigcup_{\pi,\pi'\in I}C(M(\pi'),M(\pi)),\end{align*} $$

the Conley index $CH_\ast (M(I))$ of $M(I)$ , denoted by $H_\ast (I)$ , is well defined, since $M(I)$ is an isolated invariant set for all $I\in I(<)$ . For more details, see [Reference Franzosa and Vieira8].

The simplest case of a Morse decomposition of a compact invariant set S is an attractor–repeller pair $(A, R)$ : A is an attractor in S and $R= \{x\in S \mid \omega (x) \cap A= \emptyset \}$ is its dual repeller. Note that, since S is compact the dual repeller is in fact a repeller; see [Reference Salamon20]. Then S is decomposed into $A\cup C(R,A) \cup R$ .

Given an attractor–repeller pair $(A,R)$ of an isolated invariant set S, one obtains a long exact sequence, called the attractor–repeller sequence, which relates the Conley indices of the isolated invariant sets $S, A$ and R, namely

$$ \begin{align*}\cdots \longrightarrow CH_k(A) \longrightarrow CH_k(S) \longrightarrow CH_k (R) \stackrel{\partial}{\longrightarrow} CH_{k-1}(A) \longrightarrow \cdots. \end{align*} $$

The map $\partial $ , in the previous sequence, is called the connection homomorphism or connection map. It has the property that if $\partial \neq 0$ then there exist connecting orbits from R to A in S. In many cases, it can give more information about the set of connecting orbits. For instance, if A and R are hyperbolic fixed points of indices k and $k-1$ , respectively, satisfying the transversality condition, then the connection map is equivalent to the intersection number between the stable and unstable spheres of A and R, respectively.

For a Morse decomposition $\mathcal {M}(S)$ with an admissible order $(P,<)$ , there is an attractor–repeller sequence for every adjacent pair of intervals in P. Franzosa introduced in [Reference de S. Lima, Neto, de Rezende and da Silveira11] connection matrices as devices that allow us to encode simultaneously the information in all of these sequences. Roughly speaking, connection matrices are boundary maps defined on the sum of the homology Conley indices of the Morse sets enabling each attractor–repeller sequence to be reconstructed.

More specifically, for each interval $I\subseteq P$ , set $C_\ast \Delta (I)=\bigoplus _{\pi \in I}CH_\ast (M(\pi ))$ and consider an upper triangular boundary map $\Delta (P):C_\ast \Delta (P)\rightarrow C_\ast \Delta (P)$ with respect to the partial order $<$ . Let $\Delta (I)$ be the submatrix of $\Delta (P)$ with respect to the interval I. Given an adjacent pair of intervals $I, J$ in P, one can construct the commutative diagram

where i and p are the inclusion and projection homomorphisms, respectively. In other words, we have a short exact sequence of chain complexes where the $\Delta $ s act as boundary homomorphisms. Since $(C_\ast \Delta (I),\Delta (I))$ is a chain complex, applying the homological functor H, the previous diagram produces a long exact sequence

Therefore, for every adjacent pair of intervals, the upper triangular boundary map $\Delta (P)$ generates a long exact sequence. $\Delta (P)$ is called a connection matrix if all these sequences are canonically isomorphic to the corresponding attractor–repeller sequences. In other words, for each interval I, there is an isomorphism $\phi (I):H_{\ast }\Delta (I)\rightarrow CH_{\ast }(M(I))$ such that $\phi (\{p\})=\textrm {Id}$ for every $p\in P$ , and for every adjacent pair of intervals ( $I, J$ ) the following diagram commutes:

Franzosa proved in [Reference de S. Lima, Neto, de Rezende and da Silveira11] that, given a Morse decomposition $\mathcal {M}(S)$ of an isolated invariant set S, there exists a connection matrix for $\mathcal {M}(S)$ . Moreover, he showed that non-zero entries in a connection matrix imply the existence of connecting orbits, that is, if $\Delta (p,q)\neq 0$ then $p<q$ ; in particular, for the flow defined order $<$ there is a sequence of connecting orbits from $M(q)$ to $M(p)$ .

2.2 Dynamical chain complexes

In this subsection we present some background material on dynamical chain complexes associated to Morse–Smale functions and to circle-valued Morse functions. The main references for Morse chain complexes are [Reference Banyaga and Hurtubise2, Reference Salamon21, Reference Weber22] and for Novikov complexes are [Reference Cornea and Ranicki5, Reference Pajitnov17, Reference Ranicki18].

2.2.1 Morse chain complex

A Morse–Smale function $(f,g)$ on a compact manifold $(M,\partial M)$ with boundary (possible empty) is a function $f : M \rightarrow \mathbb {R}$ together with a Riemannian metric g such that:

1. (1) the critical points are non-degenerate;

2. (2) f is regular on each boundary component N of $\partial M$ , that is, for all $x \in N$ , $ \nabla f(x) \notin T_xN \subseteq T_xM $ ;

3. (3) for any two critical points $p, q \in M$ , the stable and unstable manifolds $W^u(p)$ and $W^s(q)$ with respect to the negative gradient flow $\varphi $ of f intersect transversely.

Let $\mathrm{Crit}_{k}(f)$ be the set of critical points of f with Morse index k. Given $p \in \mathrm{Crit}_{k}(f)$ and $q\in \mathrm{Crit}_{\ell }(f)$ , define $\mathcal {M}_{pq} = W^u(p) \cap W^s(q)$ , the connecting manifold of p and q with respect to $\varphi $ , and $\mathcal {M}_p^q = W^u(p) \cap W^s(q) \cap f^{-1}(a)$ , the moduli space of p and q, that is, the space of connecting orbits from p to q, where a is some regular value of f with $f(q) < a< f(p)$ . It is well known that $\mathcal {M}_p^q$ is a $(k-\ell -1)$ -dimensional manifold. Moreover, when $\ell =k-1$ , $\mathcal {M}_p^q$ is a zero-dimensional compact manifold, hence it is a finite set.

Fix orientations of $T_p(W^u (p))$ , for all $p\in \mathrm{Crit}(f)$ . Since $W^u(p)$ is contractible, these orientations induce orientations on the tangent spaces to the whole unstable manifolds. Also, since $W^s(p)$ is contractible, it follows that the normal space $ {\mathcal V}_p W^{s}(p)$ is orientable and the orientation of $W^u(p)$ induces an obvious orientation on $ {\mathcal V}_p W^{s}(p)$ . Moreover, given $p,q\in \mathrm{Crit}(f)$ , the transversality condition implies that $T_{{\mathcal M}_{pq}}W^{u}(p)$ splits along ${\mathcal M}_{pq}$ as $ T_{{\mathcal M}_{pq}}W^{u}(p) \simeq T{\mathcal M}_{pq} \oplus {\mathcal V}_{{\mathcal M}_{pq}}W^{s}(q),$ where the last term denotes the normal bundle of $W^s(q)$ restricted to $\mathcal {M}_{pq}$ . Choose an orientation on $\mathcal {M}_{pq}$ such that this isomorphism is orientation preserving. Whenever $p\in \mathrm{Crit}_{k}(f)$ and $q\in \mathrm{Crit}_{k-1}(f)$ , the orientation on $\mathcal {M}_{pq}$ gives an orientation on the flow line associated to each $z \in \mathcal {M}_p^q$ . In this case, define $\epsilon (z):=+1$ if this orientation coincides with the one induced by the flow, otherwise define $\epsilon (z):=-1$ . Finally, let

$$ \begin{align*}n(p, q;f) = \sum_{ z\in \mathcal{M}_p^q} \epsilon(z) .\end{align*} $$

Given a Morse–Smale function $(f, g) : M \rightarrow \mathbb {R}$ , the $\mathbb {Z}$ -coefficient Morse group is the free $\mathbb {Z}$ -module $C_{\ast }(f)$ generated by the critical points of f and graded by their Morse index, that is, $C_{k}(f) = \mathbb {Z}[\mathrm{Crit}_k(f)]$ .

The $\mathbb {Z}$ -coefficient Morse boundary operator $\partial $ of f is defined on a generator p by

$$ \begin{align*} \partial_k : C_{k}(f) & \longrightarrow C_{k-1}(f) \\ p & \longmapsto \sum_{q \in \mathrm{Crit}_{k-1}(f)}n(p, q;f) q. \end{align*} $$

The pair $(C_{\ast }(f),\partial _{\ast })$ is called the Morse chain complex of the Morse–Smale function $(f,g)$ .

Salamon proved in [Reference Salamon21] that the Morse boundary operator is a special case of the connection matrix. More specifically, considering the $<_{\varphi _f}$ -ordered Morse decomposition $\mathcal {M}(M)=\{M_{\pi }\}_{\pi \in P}$ where each Morse set $M_{\pi }$ is a critical point of f and $<_{\varphi _f}$ is the flow ordering, there exists a unique connection matrix for $\mathcal {M}(M)$ , which coincides with the Morse boundary operator $\partial $ .

2.2.2 Novikov chain complex

Let $\mathbb {Z}[t,t^{-1}]$ be the Laurent polynomial ring. The Novikov ring $\mathbb {Z}((t))$ is the set consisting of all Laurent series

$$ \begin{align*}\lambda= \sum_{i\in \mathbb{Z}}a_{i} t^{i}\end{align*} $$

in one variable with coefficients $a_{i} \in \mathbb {Z}$ , such that the part of $\lambda $ with negative exponents is finite, that is, there is $n=n(\lambda )$ such that $a_k=0$ if $k<n(\lambda )$ . In fact, $\mathbb {Z}((t))$ has a natural Euclidean ring structure such that the inclusion $\mathbb {Z}[t,t^{-1}] \subseteq \mathbb {Z}((t))$ is a homomorphism.

Let M be a compact connected manifold and $f:M\rightarrow S^{1}$ be a smooth map. Given a point $x\in M$ and a neighborhood V of $f(x)$ in $S^{1}$ diffeomorphic to an open interval of $\mathbb {R}$ , the map $f|_{f^{-1}(V)}$ is identified with a smooth map from $f^{-1}(V)$ to $\mathbb {R}$ . Hence, one can define non-degenerate critical points and Morse indices in this context as in the classical case of smooth real-valued functions. A smooth map $f:M\rightarrow S^{1}$ is called a circle-valued Morse function if its critical points are non-degenerate. Denote by $\mathrm{Crit}(f)$ the set of critical points of f and by $\mathrm{Crit}_{k}(f)$ the set of critical points of f of index k.

Consider the exponential function $\mathrm{Exp}: \mathbb {R} \rightarrow S^{1}$ given by $t\mapsto \mathrm {e}^{2\pi i t}$ . The structure group of this covering is the subgroup $\mathbb {Z} \subseteq \mathbb {R}$ acting on $\mathbb {R}$ by translations. It is convenient to use the multiplicative notation for the structure group and denote by t the generator corresponding to $-1$ in the additive notation. Let $(\overline {M},p_E)$ be the infinite cyclic covering of M, where $\overline {M} = f^{\ast }(\mathbb {R}) = \{ (x,t) \in M\times \mathbb {R} \mid f(x) =[t] \in S^{1} \} $ and $p_E:\overline {M}\rightarrow M$ is induced by the map $f:M\rightarrow S^{1}$ from the universal covering $\mathrm{Exp}$ . There exists a $\mathbb {Z}$ -equivariant Morse–Smale function $F:\overline {M}\rightarrow \mathbb {R}$ which makes the following diagram commutative:

Note that if $\mathrm{Crit}(F)$ is non-empty then it has infinite cardinality. Since $\overline {M}$ is non-compact, one cannot apply the classical Morse theory to study F. To overcome this, one can restrict F to a fundamental cobordism W of $\overline {M}$ with respect to the action of $\mathbb {Z}$ . The fundamental cobordism W is defined as $ W = F^{-1}([a-1,a]), $ where a is a regular value of F. It can be viewed as the compact manifold obtained by cutting M along the submanifold $V=f^{-1}(\alpha )$ , where $\alpha = \mathrm{Exp}(a)$ . Hence, $(W, V,t^{-1}V)$ is a cobordism with both boundary components diffeomorphic to V.

From now on, we consider circle-valued Morse functions f such that the vector field $-\nabla f$ satisfies the transversality condition, that is, the lift $-\nabla F$ of $-\nabla f$ to $\overline {M}$ satisfies the classical transversality condition on the unstable and stable manifolds. Denote by $\overline {\varphi }$ the pullback of $\varphi $ , where $\varphi $ is the flow associated to $-\nabla f$ .

Fix lifts $\overline {p},\,\overline {q}\in \mathrm{Crit}(F)$ of $p, q\in \mathrm{Crit}(f)$ , respectively. Choosing arbitrary orientations for all unstable manifolds $W^{u}(p)$ of critical points of f, one considers the induced orientations on the unstable manifolds $W^{u}(t^{\ell }\overline {p})$ and $W^{u}(t^{\ell }\overline {q})$ , for $\ell \in \mathbb {Z}$ . As each path in M that originates at p lifts to a unique path in $\overline {M}$ originating at $\overline {p}$ , the space $\bigcup _{\ell \in \mathbb {Z}}\mathcal {M}({\overline {p},t^{\ell }\overline {q}})$ of flow lines of $\overline {\varphi }$ that join $\overline {p}$ to one of the points $t^{\ell }\overline {q}$ , $\ell \in \mathbb {Z}$ , is bijective with $\mathcal {M}(p,q)$ . In particular, for consecutive critical points p and q, by the equivariance of F,

$$ \begin{align*}n(t^{\ell}\overline{p},t^{\ell}\overline{q};F)=n(\overline{p},\overline{q};F)\end{align*} $$

for all $\ell \in \mathbb {Z}$ , where $n(t^{\ell }\overline {p},t^{\ell }\overline {q};F)$ is the intersection number between the critical points $t^{\ell }\overline {p}$ and $t^{\ell }\overline {q}$ of F.

Given $p \in \mathrm{Crit}_{k}(f)$ and $q \in \mathrm{Crit}_{k-1}(f)$ , the Novikov incidence coefficient between p and q is defined as

$$ \begin{align*}N(p,q;f) = \sum_{\ell \in \mathbb{Z}}n(\overline{p},t^{\ell}\overline{q};F) t^{\ell}.\end{align*} $$

For more details, see [Reference Cornea and Ranicki5, Reference Pajitnov17].

Let $\mathcal {N}_{k}$ be the $\mathbb {Z}((t))$ -module freely generated by the critical points of f of index k. Consider the kth boundary operator $\partial ^{\mathrm {Nov}}_{k}: \mathcal {N}_{k} \rightarrow \mathcal {N}_{k-1}$ which is defined on a generator $p \in \mathrm{Crit}_{k}(f)$ by

$$ \begin{align*}\partial^{\mathrm{Nov}}_{k}(p) = \sum_{q\in \mathrm{Crit}_{k-1}(f)} N(p,q;f)q\end{align*} $$

and extended to all chains by linearity. In [Reference Pajitnov17] it is proved that $\partial ^{\mathrm {Nov}}_{k}\circ \partial ^{\mathrm {Nov}}_{k+1}=0$ , hence $(\mathcal {N}_{\ast }(f),\partial ^{\mathrm {Nov}})$ is a chain complex which is called the Novikov chain complex associated to f.

4.1 p-attractor–repeller decomposition for invariant sets

It is well known that, when S is compact, each orbit has non-empty $\alpha $ - and $\omega $ -limit sets. However, this is not always the case when S is non-compact. For instance, the flow on $\mathbb {R}^2$ as in Figure 4 has a flow line $\gamma $ whose $\alpha $ - and $\omega $ -limits are empty. In this case, if we consider the usual definition of connection between two invariant sets, then in Figure 4 the orbit $\gamma $ would be a connection between R and A. In order to discard connections of these types, we restrict our analysis to the connecting orbits that have non-empty $\alpha $ - and $\omega $ -limit sets.

Figure 4

An orbit $\gamma $ with $\alpha (\gamma )=\emptyset $ and $\omega (\gamma )=\emptyset $ that is not a connection between R and A.

Recall that X is a locally compact metric space, $\varphi : \mathbb {R} \times X \rightarrow X$ is a continuous flow on X and $(\widetilde {X},p)$ is a regular cover of X, where $\widetilde {X}$ is a connected, locally path connected metric space. Let S be an invariant set in X. Given A and R invariant sets of S, the set of connections between A and R is defined by

$$ \begin{align*}C^{\ast}(R,A)=\{x \in S\backslash (A\cup R) \mid \alpha(x)\neq \emptyset, \omega(x)\neq \emptyset , \alpha(x)\subseteq R \text{ and } \omega(x)\subseteq A\}.\end{align*} $$

Given a p-attractor–repeller pair for S and $A^{\ast } := \{ x\in S \mid \omega (x) \cap A =\emptyset \}$ , note that $R = A^{\ast }$ .

It is clear that there exist invariant sets which do not admit p-attractor–repeller decompositions for any covering p.

Note that, in the previous definition, it is not required that S is compact. In this paper we are interested in an attractor–repeller decomposition such that the deck transformation group ‘acts freely and transitively’ on the attractors and repellers. This property, which is a consequence of Proposition 3.1, is stated in the next result.

It is well known that if S is a compact set and $Y\subseteq S$ , then $\omega (Y)$ and $\alpha (Y)$ are compact invariant subsets of S. Furthermore, if U is a neighborhood of $\omega (Y)$ , then there exists $t>0$ such that $Y.[t ,\infty )\subseteq U$ . A similar statement holds for $\alpha (Y)$ . Whenever S is not compact, these properties do not necessarily hold. See the example in Figure 6. However, we can still retrieve some nice properties for subsets of S which admit compact neighborhoods. The next proposition states a property of $\alpha (Y)$ and $\omega (Y)$ when S is not necessarily compact.

The following theorem is a generalization of the path lifting theorem for orbits which are contained in a compact set.

Proof. By Theorem 3.2, there are open neighborhoods $\widetilde {V}_{R}$ and $\widetilde {V}_{A}$ of $\widetilde {R} $ and $\widetilde {A} $ such that $p|_{\widetilde {V}_{R}}$ and $p|_{\widetilde {V}_{A}}$ are homeomorphisms. Let $V_{R}:= p(\widetilde {V}_{R})$ and $V_{A}:=p(\widetilde {V}_{A})$ . Since the image of $\gamma :(-\infty ,\infty )\to M$ is contained in a compact set, it follows from Lemma 4.4 that there exists $t^{\ast }$ such that $\gamma ((-\infty ,-t^{\ast }])\subset {V}_{R} $ and $\gamma ([t^{\ast },\infty ))\subset {V}_{A}$ ; see Figure 5.

Figure 5

Construction of a lift $\widetilde {\gamma }$ of the orbit $\gamma $ .

Figure 6

Example of an orbit which is not contained in a compact set.

Denote by a and b the lifts of $\gamma (-t^{\ast })$ and $\gamma (t^{\ast })$ which belong to $\widetilde {V}_{R}$ and $\widetilde {V}_{A}$ , respectively. Considering the path $\gamma {|_{{[-t^{\ast },t^{\ast }]}}}$ , by the unique path lifting property, there is a unique path $\widetilde {\beta }:[-t^{\ast },t^{\ast }]\rightarrow \widetilde {X}$ such that $\widetilde \beta (-t^{\ast })=a$ and $p\circ \widetilde \beta =\gamma $ . Since $\gamma (t^{\ast }) \in V_{A}$ , we have that $\widetilde {\beta }(t^{\ast }) \in p^{-1}(V_{A})$ . Therefore, by the transitivity and freeness of the action, there exists a unique $g \in G$ such that $\widetilde {\beta }(t^{\ast }) = gb \in g\widetilde {V}_{A}$ .

The juxtaposition $\widetilde {\gamma }$ of the paths $p^{-1}|_{\widetilde {V}_{R}}\circ \gamma |_{(-\infty , -t^{\ast }]}$ , $ \widetilde {\beta } $ and $ p^{-1}|_{g\widetilde {V}_{A}}\circ \gamma |_{ [t^{\ast },\infty )} $ is a lift of $\gamma $ such that $\alpha (\widetilde {\gamma })\subseteq \widetilde {R}$ , $\omega (\widetilde {\gamma })\subseteq g\widetilde {A}$ ; see Figure 5. The uniqueness of $\widetilde {\gamma }$ follows from the uniqueness of each one of the these paths.

The assumption that the orbit $\gamma $ is contained in a compact set is necessary in the proof of Theorem 4.5. Figure 6 shows an orbit which is not contained in any compact set. Hence, one cannot apply Theorem 4.5.

It is important to keep in mind that $C^{\ast }_g(R,A)$ , $S_{R,gA}$ and $\widetilde {S}_{\widetilde {R},g\widetilde {A}}$ depend on the choice of the sheets $\widetilde {R}$ and $\widetilde {A}$ . Note that, whenever $g\neq h$ , we have $ {S}_{{R},g{A}}\cap {S}_{{R},h{A}} = R\cup A.$ By definition, a g-orbit always has non-empty $\alpha $ - and $\omega $ -limit sets.

It is clear that an invariant set S is not necessarily evenly covered, that is, $p^{-1}(S)$ is not necessarily a union of disjoint invariant sets homeomorphic to S, even though this property holds, by assumption, for $p^{-1}(A)$ and $p^{-1}(R)$ . However, the next theorem gives a sufficient condition for ${S}_{{R},gA}$ to be evenly covered.

In order to prove Theorem 4.7, one establishes some properties of $\widetilde {S}_{\widetilde {R},g\widetilde {A}}$ in the next lemma.

The next proposition is a direct consequence of Theorems 4.7 and 3.2.

In general, $p^{-1}(S)$ is not an isolated invariant set even when S is an isolated invariant set. However, whenever S is an isolated invariant set and $\widetilde {S}_{\widetilde {R},g\widetilde {A}}$ is compact for all $g\in G$ , the next result guarantees that $p^{-1}(S)$ can be decomposed into a union of isolated invariant sets.

Proof. Since $\widetilde {S}_{\widetilde {R},g\widetilde {A}}$ is compact, ${S}_{{R},g{A}} $ is compact. Clearly ${S}_{{R},g{A}} $ is an invariant set. Hence, one needs to prove that $S_{R,gA}$ is isolated.

By Theorem 4.7, $p|_{ \widetilde {S}_{\widetilde {R},g\widetilde {A}}}: \widetilde {S}_{\widetilde {R},g\widetilde {A}}\to {S}_{{R},g{A}}$ is a homeomorphism, which can be extended to a homeomorphism $p|_{\widetilde {V}}: \widetilde {V}\to V$ where $\widetilde {V}$ is a compact neighborhood of $ \widetilde {S}_{\widetilde {R},g\widetilde {A}}$ , by Theorem 3.2.

Given $h \in G$ , let $\widetilde {U}_{h}$ be an open neighborhood of $h\widetilde {A} $ such that $\omega (\widetilde {U}_h)= h\widetilde {A}$ . Now consider the set

$$ \begin{align*}\widetilde{W} = \widetilde{V}\setminus \bigcup_{h \in G\backslash\{g\}}\widetilde{U}_h\end{align*} $$

which is still a compact neighborhood of $ \widetilde {S}_{\widetilde {R},g\widetilde {A}}$ , since no g-orbit intersects the sets $\widetilde {U}_h$ , for $h\neq g$ .

Let N be an isolating neighborhood of S. Then $N\cap p(\widetilde {W})$ is a compact neighborhood of ${S}_{R,gA}$ and ${S}_{R,gA}$ is the maximal invariant set in $N\cap p(\widetilde {W})$ . Therefore, ${S}_{R,gA}$ is an isolated invariant set.

In general, we have that $\bigcup _{g\in G}{S}_{{R},g{A}}$ is not equal to S. However, if S is a compact set (or an isolated invariant set), by Theorem 4.5 the equality holds and $p^{-1}(S)=\bigcup _{g,h \in G} h\widetilde {S}_{\widetilde {R},g\widetilde {A}} $ . Moreover, if $\widetilde {S}_{\widetilde {R},g\widetilde {A}}$ is compact, Theorem 4.10 and Proposition 4.9 guarantee that $\widetilde {S}_{\widetilde {R},g\widetilde {A}}$ is an isolated invariant set and $p^{-1}(S)$ is a disjoint union of isolated invariant sets.

In the remainder of this subsection, we establish some results for the case that $ S=\bigcup _{g\in G}{S}_{{R},g{A}}$ .

It follows from Lemma 4.11 that Theorem 4.10 holds assuming a weaker hypothesis: $\widetilde {S}_{\widetilde {R},g\widetilde {A}}$ is contained in a compact set of $\widetilde {X}$ . In particular, it holds when $\bigcup _{g\in G}\widetilde {S}_{\widetilde {R},g\widetilde {A}}$ is compact.

Considering sheets $\widetilde {R}$ over R and $\widetilde {A}$ over A, define

$$ \begin{align*}\widetilde{C}_R^S:=\bigcup_{g\in G}\widetilde{S}_{\widetilde{R},g\widetilde{A}}.\end{align*} $$

Note that $p(\widetilde {C}_R^S)=S$ and p restricted to $\widetilde {C}_R^S\setminus p^{-1}(A)$ is a homeomorphism.

The next proposition gives a condition in order to guarantee that the set of all $g\in G$ such that $C_g(R,A) \neq \emptyset $ is finite, that is, $\widetilde {C}_R^S$ can be written as a finite union of sets $\widetilde {S}_{\widetilde {R},g\widetilde {A}}$ .

In the next subsection we introduce p-connection matrices for invariant sets. Note that even when S is not an isolated invariant set but $\widetilde {S}_{\widetilde {R},g\widetilde {A}} $ is an isolated invariant set for all $g\in G$ , then $p^{-1}(S)$ can still be decomposed into a union of isolated invariant sets. We define a connecting map for this general setting. Proposition 4.9 and Theorem 4.10 guarantee that whenever S is an isolated invariant set and $\widetilde {S}_{\widetilde {R},g\widetilde {A}}$ is compact then $\widetilde {S}_{\widetilde {R},g\widetilde {A}}$ is an isolated invariant set. Hence, for this particular case, the connecting map for a p-attractor–repeller pair is well defined, as proved in the next section.

4.2 p-connection matrices for p-attractor–repeller decompositions

In this subsection we define a boundary map that ‘counts’ the flow lines between a repeller R and an attractor A by means of the lifts of these connections via the covering map p. In order to accomplish that, one needs to have at hand an algebraic structure which makes it possible to ‘count’ these flow lines in a suitable way. In what follows, we define this structure, denoted by $\mathbb {Z}((G))$ , where G is the deck transformation group associated to p.

• (H-1) If G is a finite group, we consider $\mathbb {Z}((G))$ as the group ring $\mathbb {Z}[G]$ .

• (H-2) Assume that G is a totally ordered group, that is, G is equipped with a total ordering $\leq $ that is compatible with the multiplication of G (for all $x,y,z\in G$ , $x \leq y$ implies that $zx\leq zy$ and $xz \leq yz$ ). Moreover, assume that the set $\{g\in G \mid C^{\ast }_{g}(R,A)\neq \emptyset \}$ is well ordered with respect to the order $\leq $ . For every formal series $\eta \in \mathbb {Z}[G]$ ,

  $$ \begin{align*}\eta=\sum_{g\in G}a_gg,\end{align*} $$
  where $a_g\in \mathbb {Z}$ , the support of $\eta $ is defined as
  $$ \begin{align*}\operatorname{supp}(\eta) = \{ g\in G \mid a_g\neq 0 \}.\end{align*} $$
  Let $\mathbb {Z}((G))$ be the ring of the formal series on G that have a well-ordered support. For more details, see [Reference Aparicio Monforte and Kauers1].

• (H-3) For more general G, we will assume that $\#\{g\in G;\,\, C^{\ast }_g(R,A)\neq \emptyset \}<\infty $ and $\mathbb {Z}((G))=\mathbb {Z}[G]$ .

An important particular case of (H-2) is when G is an infinite cyclic group, namely $G\,{=}\, \langle g \rangle $ . In this case, $\mathbb {Z}((G))$ is the Novikov ring $ \mathbb {Z}((t))$ , as defined in §2.2.

Note that any totally ordered group is torsion-free. The converse holds for abelian groups, that is, an abelian group admits a total ordering if and only if it is torsion-free.

The conditions on the flow in (H-2) and (H-3) are imposed to guarantee that there are no bi-infinite connections, and this fact is necessary in order to have a well-defined boundary map.

Let S be an invariant set and $(A,R)$ a p-attractor–repeller pair of S. Fix sheets $\widetilde {R}$ and $\widetilde {A}$ over R and A, respectively. Consider the subset $G'$ of G of all elements $g\in G$ such that $\widetilde {S}_{\widetilde {R},g\widetilde {A}}$ is an isolated invariant set. By Proposition 4.9, we have that $S_{{R},g{A}} = {R}\cup C^{\ast }_g({R},{A})\cup {A}$ is also an isolated invariant set.

Clearly, $(g\widetilde {A},\widetilde {R})$ is an attractor–repeller pair for $\widetilde {S}_{\widetilde {R},g\widetilde {A}}$ , for each $g\in G'$ , thus the homology Conley exact sequence of the pair $(g\widetilde {A},\widetilde {R})$ is

(1) $$ \begin{align} \cdots \stackrel{}{\longrightarrow} CH_{\ast}(g\widetilde{A}) \stackrel{i_{\ast}}{\longrightarrow} CH_{\ast}(\widetilde{S}_{\widetilde{R},g\widetilde{A}}) \stackrel{p_{\ast}}{\longrightarrow} CH_{\ast}(\widetilde{R}) \xrightarrow{\widetilde{\delta}_{\ast}(\widetilde{R}, g \widetilde{A})} CH_{\ast-1}(g\widetilde{A}) {\longrightarrow} \cdots. \end{align} $$

One can build up an analogous exact sequence for any pair $(g \widetilde {A},h\widetilde {R})$ whenever $h^{-1}g \in G'$ . By the equivariance, we have that $\widetilde {S}_{g\widetilde {R},g\widetilde {A}}\cong \widetilde {S}_{\widetilde {R},\widetilde {A}}$ and $\widetilde {S}_{h\widetilde {R},g\widetilde {A}}\cong \widetilde {S}_{\widetilde {R},h^{-1}g\widetilde {A}}$ , hence $\widetilde {\delta }_{\ast }(\widetilde {R}, \widetilde {A}) = \widetilde {\delta }_{\ast }(g \widetilde {R}, g \widetilde {A})$ and $\widetilde {\delta }_{\ast }(h\widetilde {R}, g \widetilde {A}) = \widetilde {\delta }_{\ast }( \widetilde {R}, h^{\!-\!1}g \widetilde {A})$ . Note that, if $C_g^{\ast }(R,A)$ is empty, then $\widetilde {\delta }_{\ast }(\widetilde {R}, g \widetilde {A})$ is the null map.

Fix the sets $B_k(R)=\{r_{\alpha }^k,\ \alpha \in \Lambda _k\}$ and $B_k(A)=\{a_{\alpha }^k,\ \alpha \in \Gamma _k\}$ of generators for $CH_k({R})$ and $CH_k({A})$ , respectively. Using the isomorphisms $CH_{\ast }(h\widetilde {R})= CH_{\ast }(R)$ (respectively, $CH_{\ast }(h\widetilde {A})= CH_{\ast }(A)$ ), as in Proposition 3.4, one can define

(2) $$ \begin{align} {\delta}^N_k(R,A) : \mathbb{Z}((G)) \otimes_{\mathbb{Z}[G]} \mathbb{Z}[G][B_k(R)] & \longrightarrow \mathbb{Z}((G)) \otimes_{\mathbb{Z}[G]} \mathbb{Z}[G][B_{k-1}(A)] \nonumber\\ h \otimes r_\alpha^k& \longmapsto \sum_{g\in G} h^{\!-\!1}g \otimes \widetilde{\delta}_k(h\widetilde{R}, g \widetilde{A})(r_\alpha^k) \nonumber \\ hr_{\alpha}^k & \longmapsto \sum_{g\in G}h^{\!-\!1}g \ \widetilde{\delta}_k(h\widetilde{R}, g \widetilde{A})(r_{\alpha}^k) \end{align} $$

where $\mathbb {Z}[G][B_k(R)] $ is a free $\mathbb {Z}[G]$ -module generated by $B_k(R)$ and $\widetilde {\delta }(h\widetilde {R}, g \widetilde {A})(r^k_\alpha )$ is the null map if $h^{-1}g \notin G'$ . Note that there is an injective homomorphism from $CH_k(R)$ to $\mathbb {Z}[G][B_k(R)]$ given by the map

$$ \begin{align*}\begin{array}{rcl} \xi: CH_k(R)& \longrightarrow & \mathbb{Z}[G][B_k(R)]\\ tr_\alpha^k & \longmapsto & ter_\alpha^k\end{array}\end{align*} $$

where $t\in \mathbb {Z}$ , $e\in G$ is the identity element and $r^k_\alpha \in B_k(R)$ .

Denoting $NCH_{k}(A) = \mathbb {Z}((G)) \otimes _{\mathbb {Z}[G]} \mathbb {Z}[G][B_k(A)] $ and $ NCH_{k}(R) = \mathbb {Z}((G)) \otimes _{\mathbb {Z}[G]} \mathbb {Z}[G][B_k(R)] $ , the map

$$ \begin{align*}N\Delta: NCH_{\ast}(A) \bigoplus NCH_{\ast}(R) \longrightarrow NCH_{\ast}(A) \bigoplus NCH_{\ast}(R)\end{align*} $$

defined by the matrix

$$ \begin{align*}\left( \begin{array}{@{}cc@{}} 0 & \delta^N (R,A) \\ 0 & 0 \\ \end{array} \right) \end{align*} $$

is an upper triangular boundary map called a p-connection matrix for the p-attractor–repeller decomposition of S. Denoting $NC(S) = NCH_{\ast }(A) \bigoplus NCH_{\ast }(R)$ , we have that $(NC(S),N\Delta )$ is a chain complex.

Whenever S is an isolated invariant set and $\widetilde {S}_{h\widetilde {R},g\widetilde {A}}$ is compact for all $g\in G$ , by Theorem 4.10 and Proposition 4.9, we have that $G=G'$ . Hence, the exact sequence in (1) is well defined for all $g\in G$ . Therefore $\delta ^N$ keeps track of all information on connections between adjacent invariant sets.

The next result shows that the entry $\delta ^N(R,A)$ of a p-connection matrix gives dynamical information about the connecting orbits from the repeller R to the attractor A.

Proof. Let $p_i:\widetilde {X}_{i} \rightarrow X$ be equivalent regular covering spaces of X and $G_i$ be the deck transformation groups of $p_i$ , for $i=1,2$ . Given $S \subset X$ an invariant set, a pair $(A,R)$ is a $p_1$ -attractor–repeller pair of S if and only if it is a $p_2$ -attractor–repeller pair of S.

Let $h:\widetilde {X}_2\rightarrow \widetilde {X}_1$ be a homeomorphism which provides an equivalence of the covering spaces and fix sheets $\widetilde {R},\widetilde {A}\subset \widetilde {X}_{1}$ over R and A with respect to $p_1$ . Given $g\in G_1$ , it follows from Proposition 3.7 that $h^{-1}(\widetilde {S}_{\widetilde {R},g\widetilde {A}})= h^{-1}(\widetilde {R})\cup C(h^{-1}(\widetilde {R}),h^{-1}(g\widetilde {A}))\cup h^{-1}(g\widetilde {A})$ is an isolated invariant set in $\widetilde {X}_2$ if and only if $\widetilde {S}_{\widetilde {R},g\widetilde {A}}$ is an isolated invariant set in $\widetilde {X}_1$ .

Let $\mathfrak {h}$ be the isomorphism between $G_1$ and $G_2$ induced by h. One has that $\mathfrak {h}$ induces an isomorphism

$$ \begin{align*}&H: \mathbb{Z}((G_2)) \otimes_{\mathbb{Z}[G_2]} ( \mathbb{Z}[G_2][B_k(A)] \oplus \mathbb{Z}[G_2][B_k(R)]) \\&\quad\longrightarrow \mathbb{Z}((G_1)) \otimes_{\mathbb{Z}[G_1]}(\mathbb{Z}[G_1][B_k(A)]\oplus \mathbb{Z}[G_1][B_k(R)]). \end{align*} $$

Note that H commutes with the boundary map, $H\circ \delta ^N_2=\delta ^N_1 \circ H$ , since $h_\ast \circ \widetilde {\delta }_{2\ast } =\widetilde {\delta }_{1\ast }\circ h_\ast $ , where $\widetilde {\delta }_{i\ast }$ is the connection map in the long exact sequence in (1). Then the chain complexes that arise from the covering spaces $(\widetilde {X}_1,p_1)$ and $(\widetilde {X}_2,p_2)$ are isomorphic.

In the case where S is an isolated invariant set and p is the trivial covering map, the next result proves that p-connection matrices coincide with the classical connection matrices defined by Franzosa in [Reference de S. Lima, Neto, de Rezende and da Silveira11] when we consider the homology with coefficients in a field. In this sense, the p-connection matrix theory, developed herein, generalizes the classical connection matrix theory. More specifically, when one considers the trivial covering action or one projects the group G to the trivial group ( $g\mapsto e$ ), the p-connection matrix is the usual connection matrix. In the next result, the coefficients of the homology Conley indices are assumed to be in a field.

Theorem 4.17. Let S be an isolated invariant set and $(A,R)$ a p-attractor–repeller pair of S associated to a covering map p. If $\widetilde {C}_R^S$ is compact, then the following diagram commutes:

where $\Pi $ is the following projection induced by the covering map p:

$$ \begin{align*}\begin{array}{rcl} \Pi:NCH_\ast(A)\oplus NCH_\ast (R) &\longrightarrow & CH_\ast(A)\oplus CH_\ast (R) \\ g_1\otimes a \ \oplus \ g_2\otimes r & \longmapsto & a\oplus r. \end{array} \end{align*} $$

Proof. Fix sheets $\widetilde {R}$ and $\widetilde {A}$ over R and A, respectively, and let $\widetilde {C}_R^S=\bigcup \widetilde {S}_{\widetilde {R},g\widetilde {A}}$ , where the union is over all $g\in G$ such that $ C^{\ast }(\widetilde {R},g\widetilde {A}) \neq \emptyset $ . Since $\widetilde {C}_R^S$ is compact, by Proposition 4.12 there exist $g_1,\ldots ,g_n\in G$ such that $\widetilde {C}_R^S=\bigcup _{i=1}^n \widetilde {S}_{\widetilde {R},g_i\widetilde {A}}$ .

Also $\widetilde {C}_R^S$ is an isolated invariant set. In fact, suppose that $\widetilde {C}_R^S$ is not an isolated invariant set and let N be an isolating neighborhood for S. Let $N'$ be a compact neighborhood of $\widetilde {C}_R^S$ and $N''=N'\cap p^{-1}(N)$ . Since $p^{-1}(N)$ is closed, $N''$ is a compact neighborhood of $\widetilde {C}_R^S$ . By assumption, $\widetilde {C}_R^S$ is not an isolated invariant set, hence there is $\widetilde {x}\in N''$ such that $\widetilde {x}\cdot \mathbb {R}\subseteq N''$ and $\widetilde {x}\cdot \mathbb {R} \cap \widetilde {C}_R^S=\emptyset $ . Therefore $p(\widetilde {x})\cdot \mathbb {R}\subseteq N$ and $S\subseteq p(N'')\subseteq N$ , which implies that S is not the maximal invariant set in N. This is a contradiction.

Consider the flow order $<$ for the Morse decomposition $\mathcal {M}(\widetilde {C}_R^S)=\{ g_1\widetilde {A}, \ldots , g_n\widetilde {A}, \widetilde {R}\}$ of $\widetilde {C}_R^S$ . Thus there is an index filtration $\{\widetilde {N}_0, \widetilde {N}_1, \ldots , \widetilde {N}_n, \widetilde {N}\}$ for $(\mathcal {M}(S'), <)$ such that: $(\widetilde {N}, \widetilde {N}_0)$ is a regular index pair for $\widetilde {C}_R^S$ ; $(\widetilde {N}_i, \widetilde {N}_0)$ is a regular index pair for $g_i\widetilde {A}$ ; $(\widetilde {N}, \widetilde {N}_A)$ is a regular index pair for $\widetilde {R}$ , where $\widetilde {N}_A=\bigcup _{i=0}^n \widetilde {N}_i$ ; and $\widetilde {N}_i\cap \widetilde {N}_j=\widetilde {N}_0$ for $i\neq j$ (see [Reference Franzosa and Vieira8]).

Define the functions $\tau , \tau _i:\widetilde {N} \rightarrow [0,\infty ]$ by

$$ \begin{align*}\tau(\widetilde{x})=\begin{cases} \textrm{sup}\{t>0 \ |\ \widetilde{x} \cdot [0,t]\subseteq \widetilde{N}\backslash \widetilde{N}_A\} & \text{if } \widetilde{x} \in \widetilde{N}\backslash \widetilde{N}_A,\\ 0 & \text{if } \widetilde{x} \in \widetilde{N}_A, \end{cases}\end{align*} $$

and

$$ \begin{align*}\tau_i(\widetilde{x})= \begin{cases} \textrm{sup}\{t>0 \ |\ \widetilde{x} \cdot [0,t]\subseteq \widetilde{N}\backslash \widetilde{N}_{i}\} & \text{if } \widetilde{x} \in \widetilde{N}\backslash \widetilde{N}_{i},\\ 0 & \text{if } \widetilde{x} \in \widetilde{N}_{i}. \end{cases}\end{align*} $$

Since $(\widetilde {N}, \widetilde {N}_A)$ is a regular index pair, $\tau $ is continuous. By Lemma 5.2 in [Reference Salamon20], the index pair $(\widetilde {N}, \widetilde {N}_i)$ is also regular, hence $\tau $ and $\tau _i$ are continuous.

The connection maps between index pairs $\delta : \widetilde {N}/\widetilde {N}_A \longrightarrow \Sigma \widetilde {N}_A/\widetilde {N}_0$ and $\delta _i: \widetilde {N}/\widetilde {N}_A \longrightarrow \Sigma \widetilde {N}_{i}/\widetilde {N}_{0}$ are defined as

$$ \begin{align*}\delta([\widetilde{x}])= \begin{cases} [\widetilde{x} \cdot \tau(\widetilde{x}), 1-\tau(\widetilde{x})], & 0\leq \tau(\widetilde{x})\leq 1,\\ \ [\widetilde{N}_0\times 0], & 1\leq \tau(\widetilde{x})\leq \infty, \end{cases} \end{align*} $$

$$ \begin{align*}\delta_i([\widetilde{x}])= \begin{cases} [\widetilde{x} \cdot \tau_i(\widetilde{x}), 1-\tau_i(\widetilde{x})], & 0\leq \tau_i(\widetilde{x})\leq 1,\\ \ [\widetilde{N}_0\times 0], & 1\leq \tau_i(\widetilde{x})\leq \infty. \end{cases} \end{align*} $$

When $0\leq \tau (\widetilde {x})\leq 1$ , we have that $\delta ([\widetilde {x}])=\delta _i([\widetilde {x}])$ , where i is such that $\widetilde {x}\cdot \tau (\widetilde {x})\in \widetilde {N}_{i}$ . When $\tau (\widetilde {x})\geq 1$ , we have that $\delta ([\widetilde {x}])=\delta _i([\widetilde {x}])=\delta _j([\widetilde {x}])= [\widetilde {N}_0\times 0]$ , for all i and j. Since $[\widetilde {N}_0\times 0]\in \Sigma \widetilde {N}_{i}/ \widetilde {N}_{0}$ for all i, there is a homotopy equivalence

$$ \begin{align*}\Sigma \widetilde{N}_A/\widetilde{N}_0 \simeq \bigvee_i (\Sigma \widetilde{N}_{i}/ \widetilde{N}_{0}), \end{align*} $$

where ‘ $\vee $ ’ denotes the wedge sum with base point $[\widetilde {N}_0\times 0]$ .

Note that $\delta _i(\widetilde {N}/\widetilde {N}_A)\subseteq \Sigma \widetilde {N}_{i}/ \widetilde {N}_{0}$ and if $\widetilde {x}\cdot \tau (\widetilde {x})\in \widetilde {N}_{i}\cap \widetilde {N}_{j}$ , for $i\neq j$ , then $\widetilde {x}\cdot \tau (\widetilde {x})\in \widetilde {N}_{0}$ . Therefore $\delta =\vee _i \delta _i$ . (Given maps $f:A\to B$ and $g:A\to C$ , one defines $f\vee g:A \to B\vee C$ by $f\vee g:A \stackrel {f+g}{\rightarrow } A+B\to (A+B)/{a\sim b}=A\vee B$ , where $+$ is the sum operation between topological spaces and $a\in A$ and $b\in B$ are base points.)

Applying the homological functor $\textrm {H}$ on $\Sigma ^{-1}\circ \delta =\Sigma ^{-1} \circ \vee _i\delta _i$ , we have the usual homological connection map $\delta ^N=\oplus _i \delta ^N_i$ . By projecting with respect to the covering map p, we obtain $\delta (N,N_A)=\oplus _i \delta _i(N,N_A)$ , since $\Pi $ is induced by p. Hence, the following diagram commutes:

where $(N, N_A, N_0)$ is an index filtration for $(A,R)$ .

4.3 p-Morse decomposition

Let $(P,<)$ be a partial ordered set with partial order $<$ , where P is a finite set of indices. One says that $\pi $ and $\pi '$ are adjacent elements with respect to $<$ if they are distinct and there is no element $\pi ''\in P$ satisfying $\pi ''\neq \pi ,\pi '$ and $\pi < \pi ''<\pi ' $ or $\pi ' < \pi ''<\pi $ .

In what follows, we define p-Morse decomposition for an invariant set S which is not necessarily isolated or even not compact.

Each set $M_{\pi }$ is called a p-Morse set. The partial order $<$ on P induces an obvious partial order on $\mathcal {M}(S)$ , called an admissible ordering of the p-Morse decomposition. The flow defines an admissible ordering of $\mathcal {M}(S)$ , called the flow ordering of $\mathcal {M}(S)$ , denoted $<_f$ , and such that $M_{\pi }<_f M_{\pi '} $ if and only if there exists a sequence of distinct elements of $P: \pi = \pi _0,\ldots ,\pi _n=\pi '$ , where the set of connecting orbits $C^{\ast }(M_{\pi _j},M_{\pi _{j-1}})$ between $M_{\pi _j}$ and $M_{\pi _{j-1}}$ is non-empty, for each $j = 1, \ldots ,n$ . Note that every admissible ordering of $\mathcal {M}(S)$ is an extension of $<_f$ .

Given two adjacent elements $\pi ,\pi '$ define

$$ \begin{align*}{M}_{{\pi},{\pi}^{\prime}}={M}_{\pi}\cup C^{\ast}({M}_{\pi},{M}_{\pi^{\prime}})\cup {M}_{\pi^{\prime}}\end{align*} $$

which is an invariant set. Moreover, $(M_{\pi '},M_{\pi })$ is a p-attractor–repeller pair for ${M}_{{\pi },{\pi }^{\prime }}$ . From now on fix sheets $\widetilde {M}_{\pi }$ over ${M}_{\pi }$ , for all $\pi \in P$ . Consider the subset $G_{{\pi }{\pi }^{\prime }}$ of G of all elements $g\in G$ such that

$$ \begin{align*}\widetilde{M}_{\pi,g{\pi}^\prime} :=\widetilde{M}_{\pi}\cup C^{\ast}(\widetilde{M}_{\pi},g\widetilde{M}_{\pi'})\cup g\widetilde{M}_{\pi'}\end{align*} $$

is an isolated invariant set (hence, compact). Clearly, $(g\widetilde {M}_{\pi ^{\prime }},\widetilde {M}_{\pi })$ is an attractor–repeller pair in $\widetilde {M}_{{\pi },g{\pi }^{\prime }}$ as in [Reference de S. Lima, Neto, de Rezende and da Silveira11]. Hence, for each $g\in G_{{\pi }{\pi }^{\prime }}$ there exists a long exact sequence

By Proposition 4.9, ${M}_{{\pi },g{\pi }^{\prime }}$ is an isolated invariant set. Fix a set of generators $B_k({M}_{\pi })$ for $CH_k({M}_{\pi })$ , for each $\pi \in P$ .

Denoting $NCH_{k}({M}_{\pi }) = \mathbb {Z}((G))\otimes _{\mathbb {Z}[G]} \mathbb {Z}[G][B_k(M_{\pi })] $ , let

$$ \begin{align*}N\Delta: \bigoplus_{\pi \in P} NCH_{\ast}({M}_{\pi}) \longrightarrow \bigoplus_{\pi \in P} NCH_{\ast}({M}_{\pi})\end{align*} $$

be the map defined by the upper triangular matrix

$$ \begin{align*}N\Delta = (\delta^N({\pi},{\pi^{\prime}}) )_{\pi,\pi' \in P}, \end{align*} $$

where $ \delta ^N({\pi },{\pi ^{\prime }}) $ is given by $\delta ^{N}({M}_{\pi },{M}_{\pi ^{\prime }})$ , as defined in (2), if $\pi $ and $\pi '$ are adjacent elements, and it is the null map otherwise. The possible non-zero entries of $N\Delta $ are always maps from $NCH(M_{\pi })$ to $NCH(M_{\pi '})$ , where $\pi $ and $\pi '$ are adjacent elements, and they give information on the orbits connecting $M_{\pi }$ to $M_{\pi '}$ .

The natural question herein is how to define a map from $NCH(M_{\pi })$ to $NCH(M_{\pi '})$ when $\pi $ and $\pi '$ are not adjacent elements which would give more information than the null map. This question is related to a generalization of the work in this paper to the case of a p-Morse decomposition of an invariant set. The first step in this direction is to study the behavior of the Morse sets $M(I)$ for any interval I. Since we are considering S as an invariant set, not necessarily isolated, the description of the structure of $M(I)$ is a delicate and difficult problem. For instance, $M(I)$ is not necessarily an isolated invariant set and it may not be evenly covered. We will address this problem in a future work.

However, the p-connection matrix defined herein is rich enough to describe the behavior of the connecting orbits between the Morse sets $M_{\pi }$ in the case of a p-Morse decomposition where each $M_{\pi }$ is a critical point of a circle-valued Morse function, as we prove in §5.

4.4 Examples

In this subsection we present some examples where we describe the p-connection matrix $N\Delta $ for groups G that satisfy (H-1), (H-2) and (H-3).

Example 4.19. (Klein bottle)

Let X be the Klein bottle. Consider a flow on X having one repelling singularity x, two saddle singularities $y_1,y_2$ and one attracting singularity z, as in Figure 7, where we consider the Klein bottle as the quotient space of $[0,1]\times [0,1]$ by the relations $(0, y)\sim (1, y)$ and $(x, 0) \sim (1-x, 1)$ . Consider the p-Morse decomposition where each Morse set is a singularity and the partial order is given by the flow.

Figure 7

A flow on the Klein bottle.

The universal cover of X is the plane $\mathbb {R}^2$ , and its deck transformations group G has the presentation $\langle a, b \mid ab = b^{-1}a \rangle $ . In this case, one considers $\mathbb {Z}((G))$ as the group ring $\mathbb {Z}[G]$ .

As usual in the Morse setting, one can associate the generator of the homology Conley index of each singularity with the singularity itself. With this notation and considering the unstable manifolds oriented as in Figure 7, the boundary operator is given by $\delta ^N_{2}(x,y_1) =- y_1 + b.y_1$ , $\delta ^N_{2}(x,y_2) = y_2 + bab .y_2 = y_2 + a.y_2$ , $\delta ^N_{1}(y_1,z) =b. z - a.z$ , $\delta ^N_{1}(y_2,z) =- z + b.z$ .

Example 4.20. (Double torus)

Consider a flow on the double torus X having the invariant set as in Figure 8, where we present a saddle singularity y, an attracting periodic orbit $\gamma _0$ and a repeller singularity x.

Figure 8

A flow on the double torus.

Consider the $5$ -torus $\widetilde {X}$ as a covering space of X with four leaves as in Figure 9. The deck transformation group is $G=\mathbb {Z}_2\oplus \mathbb {Z}_2=\{a,b \mid a^2=1, b^2=1\}$ , which is a finite group, hence $\mathbb {Z}((G))$ is the group ring $\mathbb {Z}[G]$ .

Figure 9

A covering space of the double torus with deck transformation group isomorphic to $\langle a,b\ |\ a^2=b^2=aba^{-1}b^{-1}=1 \rangle \cong \mathbb {Z}_2\oplus \mathbb {Z}_2$ .

As usual in the Morse setting, one can associate the generator of the homology Conley index of each singularity with the singularity itself. Let $r_1$ and $r_0$ be the generators of $CH_{i}(\gamma )$ , for $i=1$ and $i=0$ , respectively. In what follows, we compute the boundary operator $\delta ^N$ using this notation and the unstable manifolds oriented as in Figure 8. Consider the invariant set $S=\{y\}\cup C(y,\gamma )\cup \{\gamma \}$ ; the boundary operator is given by $\delta ^N_{1}(y,\gamma ) = -ar_0+abr_0 $ and $\delta ^N_{k}=0$ for $k\neq 1$ . Now consider the invariant set $S=\{x\}\cup C(x,y)\cup \{y\}$ ; the boundary operator is given by $\delta ^N_{2}(x,y) =y-ay$ and $\delta ^N_{k}=0$ for $k\neq 2$ . See [Reference Franzosa6] for more details about the computation of the boundary map $\delta $ in the presence of periodic orbits.

Example 4.21. (Solid double torus)

Consider a flow on the solid double torus M having two consecutive critical points p and q. Note that the Cayley graph of $\mathbb {Z}\ast \mathbb {Z}$ where every edge is a solid cylinder, as in Figure 10, is a universal covering of M. In this case $G=\pi _1(M)=\mathbb {Z}\ast \mathbb {Z}$ . Assume that, for each $g \in G$ there is one isolated g-orbit between p and q, and hence there are infinite isolated connections between p and q. This is the case where $\mathbb {Z}((G))$ satisfies condition (H-2) in §4.2, where $\mathbb {Z}\ast \mathbb {Z}$ is equipped with the dictionary order. Note that dynamical systems such that $\{g\in G \mid C^{\ast }_{g}(R,A)\neq \emptyset \}$ is infinite are in general not trivial to understand completely; however, the machinery constructed in this paper contributes to a better understanding of the global behavior.

Figure 10

Universal covering of the solid double torus: Cayley graph of $\mathbb {Z}\ast \mathbb {Z}$ where every edge is a solid cylinder.