On the shape Conley index theory of semiflows on complete metric spaces

In this work we develop the shape Conley index theory for local semiflows on complete metric spaces by using a weaker notion of shape index pairs. This allows us to calculate the shape index of a compact isolated invariant set $K$ by restricting the system on any closed subset that contains a local unstable manifold of $K$, and hence significantly increases the flexibility of the calculation of shape indices and Morse equations. In particular, it allows to calculate shape indices and Morse equations for an infinite dimensional system by using only the unstable manifolds of the invariant sets, without requiring the system to be two-sided on the unstable manifolds.


1.
Introduction. The Conley index theory is of crucial importance in performing the qualitative analysis of differential equations. It was first developed by C. Conley and his group for flows on locally compact spaces in 1970s [2]. Later on Rybakowski et al. extended the theory to local semiflows on complete metric spaces [14], so that it can be successfully applied to infinite dimensional dynamical systems generated by PDEs.
The basic idea of the Conley index theory is as follows. Let K be a compact isolated invariant set of a semiflow Φ on a complete metric space X. Using appropriate homotopies induced by the flow it can be shown that all the pointed quotient spaces (N/E, [E]) of index pairs (N, E) have the same homotopy type. (An index pair is a pair of suitable closed sets (N, E), where N is an isolating neighborhood of K, and E is an exit set of N .) Thus one can define the homotopy Conley index h(K) of K to be the homotopy type of the pointed space (N/E, [E]).
In general invariant sets may have very complicated topological structures. For such dynamical objects shape is another suitable concept to describe their topological quantities. It can be seen as a generalization of homotopy, and was first invented by Borsuk [1]. As spaces that have the same homotopy type have the same shape, for a compact isolated invariant set K one can immediately define the shape index s(K) as where (N, E) is a Conley index pair of K, and Sh(·) denotes the Borsuk's shape functor. Theoretically the shape Conley index works equally well as the homotopy one. In the case where X is a compact smooth manifold and Φ is a flow, Robbin and Salamon [13] introduced a certain intrinsic topology for the unstable manifold W u (K) of K. They proved that the shape index of K can be directly defined via the unstable manifold W u (K) equipped with this intrinsic topology without using index pairs. This brings us several advantages. First, it enables us to define Conley index for discrete dynamical systems. Second, it allows to calculate shape indices and Morse equations of invariant sets by using only the unstable manifolds of the invariant sets and their Morse sets. Robbin and Salamon's approach to the shape index theory was further developed in the works of Mrozek [12] and Sánchez-Gabites [15] for dynamical systems on locally compact spaces.
Situations in the case of non-locally compact spaces seem to be more complicated. In [8] Kapitanski and Rodnianski studied the shape of attractors of semiflows on complete metric spaces. They proved that the global attractor of a semiflow shares the shape of the phase space. Based on this elegant result a shape Morse theory was developed for attractors, in which the evaluation of the coefficients of Morse equations is made in terms of the unstable manifolds of Morse sets. This work was extended to isolated invariant sets of flows on locally compact metric spaces in a recent paper [16] by Sanjurjo. The author also addressed semiflows on non-locally compact spaces (see [16], Section 6). Let Φ be a given semiflow on a complete metric space X, and K be compact isolated invariant set of Φ. It was shown that if Φ is two-sided on the unstable manifold W u (K), then the shape index s(K) can be successfully calculated via its unstable manifold. Note that we do have many important examples of infinite dimensional systems arising from applications in which the corresponding semiflows are two-sided on the unstable manifolds of invariant sets (see e.g. [7,18]).
In this present work we are mainly interested in a quite general situation, in which the semiflow may fail to be two-sided on the unstable manifolds. We show that the shape index s(K) and Morse equations of an isolated invariant set K can be calculated by using either the Conley index pairs or local unstable manifolds of K and its Morse sets. In fact, we can even calculate s(K) by using a suitable closed set containing a local unstable manifold of K. Our strategy is very much like the one as we are in the situation of the classical Conley index theory.
Let Φ be an asymptotically compact local semiflow on a complete metric space X. First, instead of the usual Conley index pairs we define a new type of shape index pairs for isolated invariant sets. Roughly speaking, a shape index pair (N, E) of a compact isolated invariant set K is a pair of closed sets that enjoys the following properties: (1) N \ E is strongly admissible with E being an exit set of N ; (2) K is the maximal compact invariant set in N \ E; and (3) N \ E contains a local unstable manifold of K.
One can easily verify that each bounded Conley index pair defined as in [14] is a shape index pair defined as above. On the other hand, it is clear that shape index pairs can be constructed by using local unstable manifolds and their appropriate sections. Then using the fundamental results in [8] and [16] etc. we prove that the pointed spaces (N/E, [E]) have the same shape for all the shape index pairs (N, E) of a compact isolated invariant set K. This allows us to define the shape Conley index s(K) to be the shape Sh(N/E, [E]) of the pointed space (N/E, [E]) by using any shape index pair (N, E). Such an approach has an obvious advantage. That is, the calculation of the shape index of K can be reduced on any closed set of X that contains a local unstable manifold of K. This significantly increases the flexibility of the calculation of indices and Morse equations.
This paper is organized as follows. In Section 2 we collect some basic notions and results in the theory of topology and dynamical systems, and in Section 3 we verify the continuity property of quotient flows defined on quotient spaces of Ważewski pairs. Section 4 is the central part of this article, in which we introduce the concept of shape index pairs and define shape Conley indices for isolated invariant sets. Section 5 consists of some discussions on Morse equations. Section 6 consists of a simple illustration example.

2.
Preliminaries. In this section we collect some basic notions and results in the theory of topology and dynamical systems.
2.1. HEP and homotopy equivalence. Let X be a topological space.
Given a closed subset A of X, the pair (X, A) is said to have the homotopy extension property (HEP for short), if for any space Y and continuous map F : Let A and B be two closed subsets of X. Following Rybakowski [14] (see Chap. 1, Sec. 1.6), we define the quotient space B/A as follows.
If A = ∅, then the space B/A is obtained by collapsing A to a single point [A] in W := A ∪ B. If A = ∅, we choose any point p / ∈ B and define B/A to be the space B ∪ {p} equipped with the sum topology. In the latter case we still use the notation [A] to denote the base point p.
The result below is a pointed version of the corresponding one in [6], Pro. 0.17.
Suppose the pair (X, A) has the HEP, and that e is a strong deformation retract of A. Then (X/A, [A]) (X, e).
As a simple consequence, we have is an open subset of R + × X, and Φ enjoys the following properties: The number T x in (1) is called the maximal existence time of Φ(t, x), and D(Φ) is called the domain of Φ.
In the case when D(Φ) = R + × X, we simply call Φ a global semiflow.
Let Φ be a given local semiflow on X. For notational convenience, we will rewrite Φ(t, x) as Φ(t)x.
A subset N of X is said to be admissible (with respect to Φ), if for any sequences x n ∈ N and t n → +∞ with Φ([0, t n ])x n ⊂ N for all n, the sequence of the end points Φ(t n )x n has a convergent subsequence.
Since X may be an infinite dimensional space, to overcome the difficulty due to the lack of compactness of X, we always assume that Φ is asymptotically compact, that is, each bounded subset B of X is admissible. It is well known that this condition is naturally satisfied by many important examples from applications.

Proposition 2.3.
[10] Let x ∈ X. Then for any 0 < T < T x , there exists a δ > 0 such that T < T y for y ∈ B(x, δ). Furthermore, for any ε > 0, we have provided δ is sufficiently small.
A solution (trajectory) on an interval J ⊂ R 1 is a map γ : J → X satisfying A full solution γ is a solution defined on the whole line R 1 .

Attractors.
Let M and B be two subsets of X. We say that M attracts B, if T x = ∞ for all x ∈ B and moreover, for any ε > 0 there exists T > 0 such that A compact invariant set A ⊂ X is said to be an attractor of Φ, if it attracts a neighborhood U of itself.
Let A be an attractor. Set A Lyapunov function φ of A is said to be radially unbounded, if for any R > 0 there exists a bounded closed set B ⊂ Ω with d(B, ∂Ω) > 0 such that Proposition 2.4. A has a radially unbounded Lyapunov function φ on Ω with Proof. We infer from Li [9] that A has a radially unbounded Lyapunov function ψ on Ω. Following the procedure in [8] one can also construct a Lyapunov function ξ of A on Ω such that φ(x) ≥ d(x, A) for all x ∈ Ω. Now setting φ = ψ + ξ, we immediately obtain a Lyapunov function of A as desired.
Then F is a strong deformation retraction from U to A.

Ważewski pairs and quotient flows.
From now on we also assume that X is separable. Therefore all the quotient spaces involved in this work are completely metrizable [10]. Let A be a subset of X. For each x ∈ A, denote by t A (x) the maximal time τ such that Φ(t)x stays in A before τ , A is called strongly admissible, if it is admissible, and moreover, Φ does not In this case π is actually the identity map id N on N .) Define a quotient flow Φ of Φ on N/E as follows: Since E is N -positively invariant, it can be easily seen that Φ is well defined.
Proof. We may assume N ∩ E = ∅; otherwise N is a positively invariant set, and the proof of the lemma then becomes trivial. Let us first verify the continuity of Φ at any point (t 0 , x) ∈ R + × N/E. We split the argument into two cases.
which implies the continuity of Φ at (t 0 , x).
We may assume that r x is sufficiently small so that Set U = x∈E B(x, r x ). Then by (3.4), (3.6) and (3.7) we see that U is a neighborhood of E satisfying (3.3).
Thus by the definition of Φ we have Φ(t)y = [Φ(t)y] for all y ∈ B(x, r) and t ∈ [0, t 0 + δ], and the continuity of Φ at (t 0 , x) follows immediately from that of Φ.
(3.10) Combining this with (3.9) we deduce that . This proves what we desired in (3.8). Now let us examine the admissibility of N/E for Φ. It suffice to show that, for any sequences x n ∈ N/E and t n → ∞, the sequence Φ(t n ) x n has a convergent subsequence. We may assume Φ(t n ) x n = [E] for all n; otherwise we are done. Then for each n there exists In the following argument we denote by I(A) the maximal invariant set in A for any A ⊂ X. In general I(A) may not be compact. However, if we assume A is closed and admissible, then one can easily verify that I(A) is compact.
Proof. The proof is a slight modification of that for the same conclusion in [10], Lemma 3.7. We omit the details. 4. Shape index. In this section we introduce the notion of shape index pairs and define shape Conley indices for isolated invariant sets.

4.1.
Shape. The exposition of the basic notions and results on shape theory given here is adapted from [8,16]. For details we refer the interested reader to [1,3] and [11], etc.
We call a topological space P an absolute neighborhood retract (ANR for short) provided, for any embedding i : P → P 0 of P into a (metrizable) space P 0 , there exists a neighborhood U of i(P ) in P 0 such that i(P ) is a retract of the neighborhood U . It is known that every metric space can be embedded into an ANR as a closed subspace.
Let X and Y be two metric spaces. Suppose that X and Y are subsets of ANRs P and Q, respectively. Denote by U(X, P ) (resp. U(Y, Q)) the set of all open neighborhoods of X (resp. Y ) in P (resp. Q).
. We call f a mutation from U(X, P ) to U(Y, Q), if the following conditions are fulfilled: An example of a mutation is the trivial mutation id U(X, P ) which is comprised of the identity maps id : U → U . Q) is said to be a shape equivalence between X and Y , if there exists a mutation g : U(Y, Q) → U(X, P ) such that the compositions g • f : U(X, P ) → U(X, P ) and f • g : U(Y, Q) → U(Y, Q) are homotopic to the trivial mutations id U(X, P ) and id U(Y, Q) , respectively.
We say that X and Y are shape equivalent, if there exists a shape equivalence f between them.
Remark 4.1. That the composition g • f is homotopic to the trivial mutation id U(X, P ) (notated as g • f id U(X, P ) ) means that, whenever f : U → V ∈ f and g : V → U ∈ g, there exists U ⊂ U ∩ U so that g • f | U : U → U is homotopic to the identity map.
Remark 4.2. One may think shape as a generalization of homotopy, which can be seen from that spaces belonging to the same homotopy type have the same shape.
For pointed spaces (X, x) and (Y, y), we define a (pointed) mutation In a similar manner as in Def. 4.1, one can define shape equivalence for pointed spaces. We omit the details.
The following result is a pointed version of one of the main results in [8], and can be found in [17]. See also [4,5] for relevant results.  For a solution γ on (a, ∞) (resp. (−∞, a)), one can also define its ω-limit set ω(γ) (resp. α-limit set α(γ) ) in a similar manner. We omit the details.
Let N be a subset of X, and K ⊂ N be a compact invariant set. Define the local unstable manifold of K in N If N = X then we simply write W u N (K) as W u (K). W u (K) is called the unstable manifold of K.   [14] is a shape index pair defined as above.
We are now in position to define the shape Conley index of K via shape index pairs introduced here.  Before proving this result, let us first give some auxiliary results. Let (N, E) be a shape index pair of K, and Φ be the quotient flow on N/E defined as in Section 3.
Lemma 4.6. Φ has a global attractor A in N/E. Moreover, (4.11) Proof. We infer from Lemma 3.2 that N/E is admissible for the quotient flow Φ. Hence one can easily verify that A = ω(N/E) is compact and is precisely the global attractor of Φ. We show that (4.11) holds, thus completing the proof of the lemma. We infer from the admissibility of N/E that the maximal invariant set I(N/E) in N/E is necessarily compact. Since the global attractor of a system, if exits, is necessarily the maximal compact invariant set of the system, we deduce that A = I(N/E). Hence to prove (4.11) it suffices to check that Since K is the maximal compact invariant set of Φ in H, we find that α(γ) ⊂ K. Consequently γ(t) ∈ W u (K) for all t ≤ 0. This implies that γ(t) ∈ W u ([K]) for t ≤ 0. In particular, Proof. We first show that (4.14) Let x ∈ A. We may assume x = [E]. Then there is a full solution γ for Φ such that It is easy to see that . Thus by (4.14) one immediately concludes the validity of (4.13).
It can be assumed that K = ∅; otherwise we have W u (K) = K = ∅, which completes the proof of the lemma.
We only need to verify that W u (K) ⊂ K. Let x ∈ W u (K). Then there exists a solution γ on (−∞, 0] with γ(0) = x and γ((−∞, 0]) ⊂ W u (K). We extend γ to a solution on (−∞, T x ) (still denoted by γ), where T x is the maximal existence time of Φ(t)x. We claim that γ((−∞, T x )) ⊂ H, (4.15) where H = N \ E. Indeed, if this was not the case, then one should have t H (x) < T x . Now it is easy to see that However, this contradicts the assumption that W u N (K) ∩ E = ∅ ( as γ((−∞, t H (x)]) ⊂ W u N (K) ) and proves our claim. Because Φ does not explode in H, by (4.15) we deduce that T x = ∞. It also follows by the admissibility of H that γ is bounded on R 1 . Thus by the maximality of K in H one concludes that γ(R 1 ) ⊂ K. In particular, x = γ(0) ∈ K. Hence we see that W u (K) ⊂ K.  N, F ) is a shape index pair and has the HEP; (2) W u N (K) \ F ⊂ U ; and Proof. If N ∩ E = ∅, then N is positively invariant, and K is an attractor of Φ restricted on N . Hence W u N (K) = K. In such a case F = E fulfills all the requirements of the lemma.
Assume that N ∩ E = ∅. Let ρ be a metric on N/E such that N/E is a complete metric space. By Lemma 3.3, [E] is an attractor of Φ. We also infer from (4.14) Pick a positive number a > max x∈M φ( x) and define F = π −1 (φ a ), where π is the quotient map from W to N/E. Then F is a closed neighborhood of E in W ; see Fig. 1. We show that F satisfies (2). Indeed, we infer from M ⊂ φ a that The verification of that (N, F ) is a Ważewski pair is trivial, and we omit the details.
To complete the proof of the lemma, there remains to check the HEP of (N, F ) and the third conclusion (3).
For this purpose, we fix a number b > a and set F = π −1 (int φ b ). Then F is an open neighborhood of F in W . We claim that F is a strong deformation retract of F . Indeed, define T : φ b → R + as follows: By very standard argument (see e.g. Rybakowski [14]) it can be shown that T is continuous. Set . Then h is a strong deformation retraction from F to F , hence the claim holds true. Now it follows by Theorem 2.1 that (N, F ) has the HEP. Since π(F ) ⊂ Ω is positively invariant under the quotient flow Φ, by Pro. 2.5 [E] is a strong deformation retract of π(F ). Further by Corollary 2.1 we deduce that which completes the proof of (3).
Proof of Theorem 4.5. Let (N 1 , E 1 ) and (N 2 , E 2 ) be two shape index pairs of K.
We need to prove that . (4.16) Let N = N 1 ∩ N 2 , and E = E 1 ∪ E 2 . One can easily verify that (N, E) is a shape index pair of K. We show that hence (4.16) holds true.
where p ∈ K, from which (4.17) immediately follows.
Henceforth we assume that W u (K) = K. Then by Lemma 4.8 we deduce that In what follows we show that (4.17) holds for k = 1.
Set E u = W u N (K) ∩ E, and define Σ 1 = {y ∈ N 1 : there exist x ∈ E u and t ≥ 0 such that Φ([0, t])x ⊂ N 1 , Φ(t)x = y}; (4.18) see Figure 2. It is easy to check that E u and Σ 1 are N -positively invariant and N 1 -positively invariant, respectively. Moreover, We have Lemma 4.10. There exists a neighborhood U of K such that Σ 1 ∩ U = ∅.
Proof. We argue by contradiction and suppose the contrary. Then for each n one could find an x n ∈ E u and t n ≥ 0 with Φ([0, t n ])x n ⊂ N 1 such that Since K is compact, one can find a δ > 0 such that B(K, δ) ∩ E = ∅. Therefore it can be assumed that y n ∈ E for all n. By the definition of E u , for each n there is a solution γ n : (−∞, 0] → X contained in W u N (K) such that γ n (0) = x n . Note that γ n can be extended to a solution on (−∞, t n ] (still denoted by γ n ) by simply setting γ n (t) = Φ(t)x n . We claim that γ n ((−∞, t n ]) ∩ E 1 = ∅ for all n. Indeed, if γ n (t) ∈ E 1 for some some t ≤ t n , then by the N 1 -invariance of E 1 one should have γ n ([t, t n ]) ⊂ E 1 ⊂ E. In particular, y n = γ n (t n ) ∈ E, which leads to a contradiction and proves our claim.
Since y 0 ∈ K, we can extend γ + to a solution on [0, ∞) in H 1 by setting γ + (t) = Φ(t − τ )y 0 for t > τ . Now define Then γ is a full solution in H 1 . As K is the maximal compact invariant set in H 1 , we necessarily have x 0 = γ(0) ∈ K. However, this contradicts the fact that x 0 ∈ E.
(2) {t n } is unbounded. In this case by passing to the limit in γ n one can directly obtain a solution γ + on [0, ∞) in H 1 with γ + (0) = x 0 . Define a solution γ as in (4.21). Then γ is a full solution in H 1 , and hence x 0 = γ(0) ∈ K. This again leads to a contradiction. Now let us proceed to prove Theorem 4.5. Let U be the neighborhood of K given by Lemma 4.10. We may assume U is open. Then by Lemma 4.9 there is a closed neighborhood F of E in W = N ∪ E with K ∩ F = ∅ such that (N, F ) is a shape index pair and has the HEP; see Figure 2 where F u = W u N (K) ∩ F . Let Σ = F u ∪ Σ 1 (the grey-colored part in Fig. 3), where Σ 1 is defined as in (4.18). We claim that which completes the proof of (4.24). Now we have . Therefore by (4.23) it holds that  . Because (N, F ) has the HEP, it is easy to see that (W u N (K), F u ) has the HEP as well. Consequently we know that (W u N1 (K), Σ) has the HEP. This implies that (X 1 , q(Σ)) has the HEP. Therefore by Corollary 2.1 we have ] . (The last relation " ∼ =" in the above equation is due to that E u 1 = W u N1 (K) ∩ E 1 .) Combining this with (4.25) and (4.26), one finds that where A 1 is the global attractor of the quotient flow on N 1 /E 1 . Further by Theorem 4.2 we conclude that Likewise, it can be shown that (4.17) holds for k = 2.
Remark 4.4. Since Conley index pairs are naturally shape index pairs, one immediately concludes that shape index in the terminology here possesses continuation (homotopy) property. We omit the details.
5. Morse equations. In this section we pay some attention to Morse equations. Our results extend the corresponding ones in [8] from attractors to isolated invariant sets.

Morse decompositions of invariant sets.
For the reader's convenience, we recall briefly the definition of Morse decompositions of invariant sets.
Let K be a compact invariant set. Then the restriction Φ| K of Φ on K is a semiflow on K. A set A ⊂ K is called an attractor of Φ in K, if it is an attractor of Φ| K .
Let A be an attractor of Φ in K. Set A * is called the repeller dual to A relative to K. Accordingly, (A, A * ) is called an attractor-repeller pair in K.
Definition 5.1. An ordered collection M = {M 1 , · · · , M n } of subsets M k ⊂ K is called a Morse decomposition of K, if there exists an increasing sequence ∅ = A 0 A 1 · · · A n = K of attractors in K such that The attractor sequence of A k (k = 0, 1, · · · , n) is often called the Morse filtration of K, and each M k is called a Morse set of K.  )) (with a fixed coefficient group G) are independent of shape index pairs. This allows us to define theČech homology indexȞ * (s(K)) andČech cohomology indexȞ * (s(K))) of K, respectively, to be theČech homology theory H q (N/E, [E]) and cohomology theoryȞ q (N/E, [E]) of any shape index pair (N, E) of K.
Now suppose K has a Morse decomposition M = {M 1 , · · · , M n } with the corresponding Morse filtration ∅ = A 0 ⊂ A 1 ⊂ · · · ⊂ A n = K. Let (N, E) be a shape index pair of K, and Φ be the quotient flow on N/E defined as in Section 3. Then as in Lemma 4.6 it can be shown that M = { M 0 , M 1 , · · · , M n } forms a Morse decomposition of the global attractor A of Φ, where Here π : W := N ∪ E → N/E is the quotient map.
Let φ be a radially unbounded strict Morse-Lyapunov function of A corresponding to the Morse decomposition M , and let a k = φ( M k ) (k = 0, 1, · · · , n). Then a 0 < a 1 < · · · < a n . For each k ≥ 0 we fix a number b k with a k < b k < a k+1 (we assign a n+1 = ∞). Set N k = φ b k . Clearly N 0 ⊂ N 1 ⊂ · · · ⊂ N n . It is easy to verify that ( N k , N k−1 ) is a shape index pair of M k for each k ≥ 1. Consequently (N k , N k−1 ) is a shape index pair of M k , where N k = π −1 ( N k ), k = 0, 1, · · · , n.
Similarly we also know that (N k , N 0 ) is a shape index pair of A k for k ≥ 1. Since N 0 ⊂ N 1 ⊂ · · · ⊂ N n , by very standard argument (see e.g. [16]) one can obtain the following Morse equation where and ∂ q,k is the boundary operator fromȞ q (N k , N k−1 ) toȞ q−1 (N k−1 , N 0 ). Here we have assumed that all the relative homology groups have finite ranks (results in this line can be found in [8,13] etc.). As (N k , N k−1 ) and (N k , N 0 ) are shape index pairs of M k and A k , respectively, we haveȞ * (N k , N k−1 ) =Ȟ * (s(M k )),Ȟ * (N k , N 0 ) =Ȟ * (s(A k )). Hence (5.27) can be rewritten as follows: n k=1 ∞ q=0 t q rankȞ q (s(M k )) = ∞ q=0 t q rankȞ q (s(K)) + (1 + t)Q(t). Remark 5.6. Similar results remain valid forČech cohomologies. We omit the details.
6. An example. In this section we give an easy example for the computation of shape indices, which may help the reader to have a better understanding to the concept of shape index pairs introduced here. Consider the parabolic problem: in Ω, where Ω is bounded domain in R m , and It is well known (see e.g. [18]) that for each u 0 ∈ H = L 2 (Ω), the problem (6.31) has a unique solution u ∈ C(R + ; L 2 (Ω)) with u ∈ L 2 (0, T ; H 1 0 (Ω)) ∩ L 2p (0, T ; L 2p (Ω)), ∀ T > 0.
Moreover, the solution operator u 0 → u(t) generates a global semiflow Φ on L 2 (Ω). Φ is asymptotically compact and has a global attractor A in L 2 (Ω). If the system has only a finite number of equilibria: e 1 , e 2 , · · · , e n , then M = {e 1 , e 2 , · · · , e n } forms a Morse decomposition of A. Suppose that we want to write out explicitly the Morse equation of A. Then one has to make out all the homology groupš H q (s(e k )) andȞ q (s(A)). As A is the global attractor, we infer from [8] that s(A) = Sh(H, p) = Sh({p}, p).
(The second equality in the above equation is due to the fact that H is contractible.) Hence we find thatȞ q (s(A)) = 0. Now let us try to calculateȞ q (s(e k )). If e k is hyperbolic, thenȞ q (s(e k )) is completely determined by the local unstable manifold of e k , and the calculation of H q (s(e k )) is somewhat trivial. The situation in the case when e k is not hyperbolic seems to be complicated. In such a case the local unstable manifolds of e k usually remain unknown. Noticing that the system has a natural Morse-Lyapunov function To overcome this difficulty, the usual way was to restrict the system on the attractor A and think of A as the phase space. (It can be shown that A is a compacta in V .) But to do so, one first need to calculate the attractor A. Another drawback is that the continuity of the semigroup Φ in the topology of V can be hardly examined. Here we advocate to use shape index pairs defined as in Section 4.
As J β α = J β \ J α is a bounded subset of H, it follows by the asymptotic compactness of Φ that J β α is strongly admissible. (Recall that Φ is a global semiflow, hence no solutions explode.) Since both J α and J β are positively invariant for Φ, clearly J a is J β -positively invariant. The verification of that J β α contains a local unstable manifold of e k and that J α is an exit set of J β is also trivial. We omit the details. Hence we see that (J β , J α ) is indeed a shape index pair. Remark 6.7. Note that the pair (J β , J α ) in the above argument may fail to be a Conley index pair of e k , because J β α is in general not a neighborhood of e k .