CHAPTER 5 PROOFS OF THE CONTINUATION RESULTS

This chapter provides proofs of Theorems 3.1–3.3, yielding the invariance of intersection numbers under continuation, and of Theorem 3.7, showing that a path in a space of isolated invariant sets can be lifted to the corresponding space of Čech homology classes of the isolated invariant sets if the initial point of the lift is an intersection class of homology classes in the Conley indices of the invariant set below it. These proofs are inextricably bound up with the definition of the map between Conley indices defined by a path in a space of isolated invariant sets as given in [K3]. Thus, in preparation for the proofs we (i) review the notion of a morphism between connected simple systems (maps between Conley indices are a special case) which leads to the notion of a category of connected simple systems that are subcategories of a category H, denoted by CSS(H), (ii) review how a path β in a space of isolated invariant sets S(φ) defines maps between Conley indices

C (S_{β}(0)) ⟶^{C(β)} C (S_{β}(1)) and C∗(S_{β}(0)) ⟶^{C∗(β)} C∗(S_{β}(1))

both of which are equivalences in CSS(T^{∗′}), (iii) show that the singular homology functor can be extended to a functor on CSS(T^{∗′}) to G_{R}M whose application to C (β) and C∗(β) yields the graded R-module isomorphisms

H̃_{∗}C (S_{β}(0)) ⟶^{C(β)_{∗}} H̃_{∗}C (S_{β}(1)) and H̃_{∗}C∗(S_{β}(0)) ⟶^{C∗(β)_{∗}} H̃_{∗}C∗(S_{β}(1))

used to state the continuation results of Chapter 3. To facilitate the exposition, the preparatory material will be given in the order (i), (iii), (ii).

A. Maps between Conley Indices from Paths of Invariant Sets. As shown in [K3], the collection of connected simple systems that are subcategories of some category H is itself the class of objects of a category, denoted CSS(H) and called the category of connected simple systems that are subcategories of H. A morphism in this new category is called a map of connected simple systems. Formally, as defined in [K3], a map of connected simple systems F: A → B, with A and B both subcategories of H, is a covariant functor on the product category A × B to the morphism category of H satisfying certain additional properties. This formal definition and the definition of the composition of two maps of connected simple systems as given in [K3] together are rather long and are not really needed for the proofs of the continuation results; hence, only an informal characterization of the complete definitions will be given.

5.1 Definition. (A) Informally, if A and B are connected simple systems in CSS(H), a map of connected simple systems F: A → B is a collection of commutative diagrams of morphisms in H of the form

(5.1)  \begin{matrix}A & ⟶^{F_{A}^{B}} & B \\ _{h_{A}^{A A′}} ↓ & & ↓_{h_{B}^{B B′}} \\ A′ & ⟶_{F_{A′}^{B′}} & B′ \end{matrix}

where A and A′ are objects of A and B and B′ are objects of B. As the notation in (5.1) implies, for any two such pairs of objects (A, B) and (A′, B′), there must be one, and only one, such diagram in the collection. Note that this uniqueness implies that the collection is completely specified if F_{A}^{B} is given for a single pair of objects (A, B). For necessarily,

(5.2)  F_{A′}^{B′} = h_{B}^{B B′} ^{∘} F_{A}^{B} ^{∘} h_{A}^{A′ A} .

(B) In particular, for any connected simple system A in CSS(H), the identity morphism of A in CSS(H), denoted 1_{A}, is specified by (1_{A})_{A}^{A} = 1_{A} for any A ∈ ob(A). It follows from (5.2) and the connected simple system identities (1.1) that if A′ and A′′ are any objects of A, then (1_{A})_{A′}^{A′′} = h_{A}^{A′ A′′}.

(C) If G: B → C is also a map of connected simple systems in CSS(H), then informally the composition G ^{∘} F is obtained by juxtaposing a diagram in G to the right of a diagram in F whenever the right vertical arrow of the diagram in F coincides with the left vertical arrow of the diagram in G and then deleting this common arrow from the juxtaposed diagrams. Some work must be done to show the resulting collection of diagrams has the necessary uniqueness property. See [K3] for details. Clearly this composition is then associative.

It is easily verified that with this definition of composition, the self-map of each connected simple system defined in (B) indeed functions as an identity morphism. Hence, (A), (B), and (C) together yield that CSS(H) is a well-defined category.

5.1.1 Remark. From (5.2), (B), and (C) it follows that a map of connected simple systems F: A → B in CSS(H) is an equivalence, i.e., there exists G: B → A in CSS(H) so that G^{∘} F = 1_{A} and F ^{∘} G = 1_{B}, if, and only if, F_{A}^{B} is an equivalence in H for one, hence every, pair (A, B) ∈ ob(A) × ob(B).

Recall that in Definition 1.5 we observed that every connected simple system is a directed set and used this to show that to every functor on T^{∗′}, the homotopy category of pointed topological spaces, into _{R}M (or G_{R}M, or ∂_{R}M′) one could assign limit modules to a connected simple system that is a subcategory of T^{∗′}. The following result completes that construction in that it shows that each such functor induces a functor on CSS(T^{∗′}) to _{R}M (or G_{R}M, or ∂_{R}M′) and generalizes it by replacing T^{∗′} with any category H. In particular, for C a connected simple system and a subcategory of a category H and F a functor on H to a category X,

(5.3)  {{F(X)}_{X∈ob (C)}, {F(h_{C}^{X X′}}_{X⪯X′∈ob (C)}}

is as F is covariant or contravariant a direct or inverse system of objects and isomorphisms in X directed over ob(C). The limits taken in the following theorem are those of the system (5.3).

5.2 Theorem. Associated to each functor F on a category H to X where X is one of the categories _{R}M, G_{R}M, or ∂_{R}M′ is an induced functor of the same variance also denoted by F on CSS(H) to X with the following properties:

(1) If F is covariant, F(C) := \underrightarrow{lim} F(X) and for each X ∈ ob(C), the canonical map ι^{X}: F(X) → F(C) is an X -isomorphism;

(2) if F is contravariant, F(C) := \underleftarrow{lim} F(X) and for each X ∈ ob(C), the canonical map π_{X}: F(C) → F(X) is an X -isomorphism;

(3) if f: A → B is a map of connected simple systems in CSS(H), then F(f) is a morphism in X with the property that

(i) if F is covariant there is a commutative diagram

\begin{matrix}F(A) & ⟶^{F(f)} & F(B) \\ _{i^{A}}↑ & & ↑_{i^{B}} \\ F(A) & ⟶^{F(f_{A}^{B})} & F(B) \end{matrix} for A ∈ ob(A), B ∈ ob(B);

(ii) if F is contravariant there is a commutative diagram

\begin{matrix}F(A) & ⟵^{F(f)} & F(B) \\ ^{π_{A}}↓ & & ↓^{π_{B}} \\ F(A) & ⟵^{F(f_{A}^{B})} & F(B) \end{matrix} for A ∈ ob(A), B ∈ ob(B).

Proof. Parts (1) and (2) of the theorem follow from the constructions given in Definition 1.5(A) by substituting the objects and morphisms of H for those of T^{∗′} as no categorical properties of T^{∗′} specific to it were used in the construction. Note, however, that F here corresponds to F̃ in the construction of Definition 1.5(A) and that there is no correspondent to the functor F in that definition. It also follows from the observations made in Definition 1.5(A) that, for any C ∈ ob(CSS(H)), for any X ∈ ob(C), when F is covariant the canonical map ι^{X} is an X -isomorphism, but when F is contravariant the canonical map π_{X} is an X -isomorphism.

To avoid confusion in giving the definition of how the induced functor acts on maps of connected simple systems, let us temporarily use F̄ to denote the induced functor.

For each f: A → B a map of connected simple systems in CSS(H), the morphism F̄(f) is defined as follows: Choose A ∈ ob(A) and B ∈ ob(B) and if F is covariant set

F̄(f) := i^{B} ^{∘} F(f_{A}^{B})^{∘} (i^{A})^{−1}: F̄(A) → F̄(B) ,

however, if F is contravariant, set

F̄(f) := (π_{A})^{−1} ^{∘} F(f_{A}^{B})^{∘} π_{B}: F̄(B) → F̄(A) .

It is immediate from the definition of f as a collection of commutative diagrams of the same type as in (5.1) that the definition of F̄(f) is independent of the choice of A and B. It is then easily verified that so defined F̄ is a functor on CSS(H) to X of the same variance as F. □

From now until the proof of Theorem 3.1 is completed, in the notation of Chapter 1, let φ: G_{Λ} → M denote a continuous family of C^{s} flows in M (0 ≤ s ≤ ∞) and let π_{φ}: S(φ) → Λ be the natural projection on the associated space of isolated invariant sets.

Before we can define the map of connected simple systems induced by a path in S(φ) between the Conley indices at the endpoints of the path, we must first define a class of maps between Conley indices of isolated invariant sets where one invariant set is a slice of the other and so require the following preliminary definition and lemma.

5.3 Definition. Let τ: M × R × Λ → M × Λ × R be the homeomorphism transposing the second and third coordinates, and for K a compact subset of Λ, set G_{K} equal to G_{Λ} ∩ (M × R × K) and let τG_{K} be the image of G_{K} under τ. Then define φ_{K}: τG_{K} → M × K by φ_{K}((u, λ), t) := (u ·^{λ}t, λ). The definition of a C^{0} flow on a manifold made in Chapter 1 works equally well if “manifold” is replaced by “locally compact Hausdorff space”, and so defined, it follows that φ_{K} is a continuous flow in the locally compact Hausdorff space M × K because φ is a continuous family of flows in M.

Let us also note that the definitions of “isolated invariant set” and “Conley index” given in Chapter 1 also carry over to the setting of a continuous flow in a locally compact Hausdorff space, which in fact is the setting used in [C]. However, in that setting, see [K5] as regards the definition of the morphisms in a Conley index given in Definition 1.3(C).

The following result is due to Conley; for a proof see [C, Theorem IV.2.1.B] or [Sl, Lemma 6.4(i)].

5.4 Lemma. For K a compact subset of Λ, S_{K} is an isolated invariant set of φ_{K} if, and only if, there exists a continuous section σ: K → S(φ) of π_{φ} with the property

S_{K} = ⋃_{λ∈K} S_{σ}(λ) × {λ} .

As a consequence of this lemma we can make the following definition.

5.5 Definition. For compact K ⊂ M and S_{K} an isolated invariant set of the flow φ_{K}, let σ be the section of π_{φ} guaranteed by Lemma 5.4 and for each λ ∈ K define a map of connected simple systems F (λ, K): C (S_{σ}(λ)) → C (S_{K}) as follows. Choose an index pair ⟨N_{1}, N_{0}⟩ for S_{K} and for each λ ∈ K, for i = 0, 1, define N_{i}(λ) by the equality

N_{i}(λ) × {λ} = N_{i} ∩ (M × {λ}) .

So defined, it is easy to verify that ⟨N_{1}(λ), N_{0}(λ)⟩ is an index pair for S_{σ}(λ) for each λ ∈ K. Hence, for each λ ∈ K the embedding ι_{λ K}: M → M × K given by ι_{λ K}(u) = (u, λ) induces a continuous map of quotients

ῑ_{λ K}: N_{1}(λ)/N_{0}(λ) → N_{1}/N_{0}

and via formula (5.2) applied to the situation at hand thereby defines a map of connected simple systems from C (S_{σ}(λ)) to C (S_{K}) which we take to be F (λ, K).

N.B. It is quite important to the proof of the invariance of intersection numbers under continuation given below that for each λ ∈ K, the map of connected simple simple systems F (λ, K) as just defined is independent of the index pair ⟨N_{1}, N_{0}⟩ chosen and used to make the definition. See [K3, Proposition 2.5 and Corollary 2.22] or [Sl, Proposition 6.5] for a proof of this fact.

The definition of C (β) rests heavily on the fact that “small” compact K can be found for which F (λ, K) is an equivalence. Note that by Remark 5.1.1, F (λ, K) will be an equivalence in CSS(T^{∗′}) if, and only if, ῑ_{λ K} is a homotopy equivalence, and for some compact K ⊂ Λ, this is indeed the case. Specifically, the following existence lemma is proved in [C,Theorem IV.2.2C] given W open in Λ and σ: W → S(φ) a local section of π_{φ}.

5.6 Lemma. For each ν ∈ W given N_{ν} that ν-isolates S_{σ}(ν), there exists V_{ν} open, ν ∈ V_{ν} ⊂ W ∩Λ(N_{ν}), with the property that if ν ∈ K ⊂ V_{ν} and K is compact and contractible, then F (λ, K): C (S_{σ}(λ)) → C (S_{K}) is an equivalence in CSS(T^{∗′}).

5.7 Definition. Suppose that β is a path in S(φ). Let us first define C (β) in the following special case.

(A) Assume Λ = [0, 1] and β is a section of π_{φ}. For each ν ∈ Λ choose N_{ν} that ν-isolates S_{β}(ν) and apply Lemma 5.6 to get an open cover {V_{ν} : ν ∈ Λ} of [0, 1] so that each element V_{λ} of this cover satisfies λ ∈ V_{λ} ⊂ Λ(N_{λ}) and has the property that if λ ∈ K ⊂ V_{λ} and K is compact and contractible, then F (λ, K) is an equivalence.

From the compactness of [0, 1] it follows that there exists a partition of [0, 1], P: 0 = λ_{0} < … < λ_{n} = 1, subordinate to the cover {V_{ν} : ν ∈ Λ}. Set K_{j} := [λ_{j−1}, λ_{j}] and define

S_{j} := S_{K_{j}} := ⋃_{λ∈K_{j}} S_{β}(λ) × {λ} (j = 1, …, n)

so that S_{j} is an isolated invariant set of the flow φ_{K_{j}} as a consequence of Lemma 5.4. Then the map of Conley indices

F_{i j} := F (λ_{i}, K_{j}): C (S_{σ}(λ_{i})) → C (S_{j}) (i = j − 1, j; j = 1, …, n)

is an equivalence in CSS(T^{∗′}) by virtue of which there is a well-defined equivalence F_{j j}^{−1}^{∘}F_{j−1 j}: C (S(λ_{j−1})) → C (S(λ_{j})) for j = 1, …, n. Thus, there is a well-defined equivalence in CSS(T^{∗′})

F (β, P): C (S_{β}(0)) → C (S_{β}(1))

given by the composite

(5.4)  F (β, P) := (F_{n n}^{−1} ^{∘} F_{n−1 n})^{∘} ⋯^{∘} (F_{1 1}^{−1} ^{∘} F_{0 1} ) ,

and we define

C (β) := F (β, P) .

This definition of C (β) is in fact independent of (i) for ν ∈ Λ, the choice of isolating neighborhood N_{ν} that ν-isolates S_{β}(ν), (ii) the choice of cover {V_{ν} : ν ∈ Λ}, and (iii) the choice of partition P subordinate to that cover so long as F_{i j} (i = j − 1, j; j = 1, …, n) is an equivalence; see [C, §IV.2.3] and [K3,Proposition 2.11].

(B) Let us now treat the general case: Λ is some locally compact Hausdorff space and β is an arbitrary path in S(φ). In essence, we reduce this case to the previous case by pulling back π_{φ} by λ_{β}. Specifically, set Λ′ = [0, 1] and without loss of generality we can assume Λ′ and Λ are disjoint; otherwise identify Λ′ with a copy of itself that is disjoint. For each λ′ ∈ Λ′, set G_{λ′} := G_{λ_{β}(λ′)} and define a C^{s} flow ψ_{λ′} in M with domain G_{λ′} by

ψ_{λ′}(u, t) := φ_{λ_{β}(λ′)}(u, t) .

Then {ψ_{λ′}}_{λ′∈Λ′} is a continuous family of C^{s} flows in M because {φ_{λ}}_{λ∈Λ} is. By definition, ψ = λ_{β}^{∗}φ; i.e., ψ(u, t, λ′) = φ(u, t, λ_{β}(λ′)}. Note that S is isolated with respect to the flow ψ_{λ′} if, and only if, S is isolated with respect to the flow φ_{λ_{β}(λ′)}. Define a path β′ in S(ψ) by β′(λ′) = (S_{β}(λ′), λ′) and note that β′ is a section of S(λ_{β}^{∗}φ), whence C (β′) has already been defined by part (A) of this definition. Then define C (β) := C (β′). Since by definition S_{β′}(λ′) = S_{β}(λ′) we get that C (β) is indeed a map between C (S_{β}(0)) and C (S_{β}(1)).

The definition of C (β) just given is equivalent to the definition given in [K3]^{1} although the formalism used to describe continuous families of flows in [K3] follows that of [C] and is based on the idea of embedding each flow in M into the space of regular curves in M × Λ making the notation used there much more complicated. It is also shown in [K3] that C (β) is actually an invariant of the homotopy class of a path β (endpoints fixed) and in fact is a contravariant functor on Π(S(φ)), the fundamental groupoid of S(φ), to CSS(T^{∗′}); see [K3, Theorem 2.19 and Corollary 2.22]. (The fundamental groupoid of a space X is the small category Π(X) with X the set of objects and for points x, x′ ∈ X, the set of morphisms Π(x, x′) is the set of homotopy classes of paths (endpoints fixed) in X from x′ to x with composition the usual multiplication of paths.)

(C) Similarly there is a contravariant functor C∗: Π(S(φ)) → CSS(T^{∗′}) that assigns the reverse time Conley index C∗(S) to a point (S, λ) ∈ S(φ) and assigns to a path class β in S(φ) a map of connected simple systems C∗(β) : C∗(S_{β}(0)) → C∗(S_{β}(1)) where C∗(β) is defined just as C (β) is, but using the family of timereversed flows {φ∗_{λ}}_{λ∈Λ}; i.e., index pairs relative to the family of time-reversed flows are used in the definition rather than index pairs relative to the given flows.

(D) As noted at the beginning of the chapter, for β a path or the homotopy class of a path in S(φ), the graded R-module homomorphisms (coefficients taken in an R-module G)

H̃_{∗}C (S_{β}(0)) ⟶^{C(β)_{∗}} H̃_{∗}C (S_{β}(1)) and H̃_{∗}C∗(S_{β}(0)) ⟶^{C∗(β)_{∗}} H̃_{∗}C∗(S_{β}(1))

1However, there is an error in [K3] in the definition of the pullback of a product parameterization. Part (2) of that definition beginning on line 6 of page 305 should be replaced with the following:

(2) Define f^{∗}Ψ to be the subset of Ψ_{f(Λ′′)} × Λ′′ consisting of those points (γ′, λ′′) satisfying

γ′ ∈ Ψ_{f(λ′′)}.

are obtained by applying Theorem 5.2 to the reduced singular homology functor (coefficients in G) and the maps of connected simple systems C (β) and C∗(β). Because the path class β regarded as a morphism of Π(S(φ)) is invertible with the inverse being given by running the path backwards, the fact that C and C∗ are functors on Π(S(φ)) implies that C (β) and C∗(β) are invertible, hence so too are C (β)_{∗} and C∗(β)_{∗}.

B. The Proof of Theorems 3.1, 3.2, 3.3, and 3.7.

5.8 Proof of Theorem 3.3. For each λ ∈ [0, 1], choose C^{r} isolating blocks with corners N_{λ} and N∗_{λ} for S_{σ(λ)} as guaranteed by Proposition 4.4 so that the pair of index pairs ((N_{λ}, N_{λ}^{+}), (N∗_{λ}, N_{λ}^{∗ −})) is AIP(regular) for S_{σ(λ)}. Because of the continuity of the curve of isolated invariant sets, the transversality of the vectorfield X_{λ} to the codimension one submanifolds containing the exit and entrance sets of N_{λ} and N∗_{λ}, and the continuity of λ → X_{λ}, about each λ ∈ [0, 1] is centered an interval J_{λ} so that λ′ ∈ J_{λ} implies ((N_{λ}, N_{λ}^{+}), (N∗_{λ}, N_{λ}^{∗ −})) is AIP(regular) for S_{σ}(λ′).

Choose a partition of [0, 1], P : 0 = λ_{0} < …, λ_{n} = 1, subordinate to the cover of [0, 1] by the intervals J_{λ} (λ ∈ [0, 1]). Let K_{j} and S_{j} (j = 1, …, n) be defined relative to the partition P as in part (A) of Definition 5.7. Because K_{j} is contractible, it follows that for each λ ∈ K_{j}, the embedding of pairs

ι_{λ j}: (N_{λ_{j}}, N_{λ_{j}}^{+}) → (N_{λ_{j}} × K_{j}, N_{λ_{j}}^{+} × K_{j})
(5.5)  u ↦ (u, λ)

induces a homotopy equivalence ῑ_{λ j}: N_{λ_{j}}/N_{λ_{j}}^{+} → (N_{λ_{j}} × K_{j})/(N_{λ_{j}}^{+} × K_{j}) between index spaces and thereby determines the equivalence of connected simple systems F (λ, K_{j}): C (S_{σ}(λ)) → C (S_{j}). Thus with F_{i j} := F (λ_{i}, K_{j}) for i = j − 1, j and j = 1, …, n,

C (β) = (F_{n n}^{−1} ^{∘} F_{n−1 n})^{∘} ⋯^{∘} (F_{1 1}^{−1} ^{∘} F_{0 1} ) .

Analogously, the embedding of pairs

ȷ_{λ i}: (N∗_{λ_{i}}, N_{λ_{i}}^{∗ −}) → (N∗_{λ_{i}} × K_{i}, N_{λ_{i}}^{∗ −} × K_{i})
(5.6)  u ↦ (u, λ)

induces a homotopy equivalence j̄_{λ i}: N∗_{λ_{i}}/N_{λ_{i}}^{∗ −} → (N∗_{λ_{i}} × K_{i})/(N_{λ_{i}}^{∗ −} × K_{i}) of index spaces for the reverse time flow that in turn determines an equivalence of connected simple systems G(λ, K_{i}): C∗(S_{σ}(λ)) → C∗(S_{i}). Thus with G_{ℓ i} := G(λ_{ℓ}, K_{i}) for ℓ = i − 1, i and i = 1, …, n,

C∗(β) = (G_{n n}^{−1} ^{∘} G_{n−1 n})^{∘} ⋯^{∘} (G_{1 1}^{−1} ^{∘} G_{0 1} ) .

Let us note that the rule r(u, λ) = u defines a left inverse to the embedding (5.5); hence induces a map on quotients r̄ that is a homotopy inverse to ῑ_{λ j}; hence determines the map of connected simple systems F (λ, K_{j})^{−1}. In the same way the rule r induces a map on quotients that is a homotopy inverse to j̄_{λ i} and determines G(λ, K_{i})^{−1}.

Set α_{0} := α and γ_{0} := γ and inductively define α_{j} := F_{j j ∗}^{−1}F_{j−1 j ∗}α_{j−1} and γ_{j} := G_{j j ∗}^{−1}G_{j−1 j ∗}γ_{j−1} for j = 1, …, n. To finish, it suffices to show that

(5.7)  ^{#}(α_{j−1} ⌢_{•} γ_{j−1}) = ^{#}(α_{j} ⌢_{•} γ_{j})  (j = 1, …, n) .

To show that (5.7) holds, let us first make the simple observation that if V ⊂ M and ξ ∈ H_{0}(V) and i_{M}^{V}: H_{∗}(V) → H_{∗}(M) is the inclusion induced homomorphism, then

⟨1, i_{M}^{V}ξ⟩ = ⟨1, ξ⟩

where on the left 1 ∈ H^{0}(M), but on the right 1 ∈ H^{0}(V) and is in both cases the cohomology class of the augmentation on 0-chains into R. Hence for each compact K ⊂ M, for each ξ̌ ∈ Ȟ_{0}(K), and for each V ∈ N (K),

(5.8)  ⟨1, ξ̌_{M}⟩ = ⟨1, ξ̌_{V} ⟩ .

Suppose then that α is a relative class in N_{λ_{j}} modulo N_{λ_{j}}^{+}) representing α_{j−1}. Because the homotopy class ῑ_{λ_{j} j}^{−1} ^{∘} ῑ_{λ_{j−1} j} is represented by r̄^{∘} ῑ_{λ_{j−1} j} = 1_{N_{λ_{j}} /N_{λ_{j}}^{+}}

it follows that the map of connected simple systems F_{j j}^{−1} ^{∘} F_{j−1 j} is determined by the identity map 1_{N_{λ_{j}} /N_{λ_{j}}^{+}} . It is then immediate that α also represents α_{j}.

Similarly, if γ is a relative class in N∗_{λ_{j}} modulo N_{λ_{j}}^{∗ −}) representing γ_{j−1}, then

also γ represents γ_{j}. Choose V ∈ M_{Δ̸ o}^{(2)}(N_{λ_{j}}). Then from Definition 2.12, from Theorem 2.11 (in particular Remark 2.11.1 with P := N_{λ_{j}}), from the cofinality of

{N_{λ_{j}}} in M_{Δ̸ c}^{(2)}(N_{λ_{j}}), from the definition of the pairing 𝔏_{N_{λ_{j}}}^{N_{λ_{j}}} (a particular case of

(2.8)), and from the cofinality of {V_{1} ∩ V ∗_{1}} in N (V_{1} ∩ V ∗_{1}), it follows that

(5.9)  (α_{k} ⌢_{•} γ_{k})_{V_{1}∩V ∗_{1}} = 𝔏_{V_{1}∩V ∗_{1}}^{N_{λ_{j}}}(α ⊗ γ)  for k = j − 1, j.

It is immediate from (5.9) and the observation (5.8) that (5.7) holds as desired. □

To prove the analogous result without the smoothness assumption on M and the flows φ_{λ} (λ ∈ Λ) requires a technical result that describes how certain pairs of index pairs that are AIP(thick) for S_{σ}(λ) can be perturbed to index pairs that are AIP(thick) for S_{σ}(λ′) for λ′ sufficiently close to λ. This will allow us to replace the use of 1_{N_{λ_{j}} /N_{λ_{j}}^{+}} in the preceding demonstration by a suitable inclusion induced

map of index spaces, but otherwise the proof is essentially unchanged. The proof of the following lemma is deferred until after the proof of Theorem 3.1.

5.9 Lemma. Given a local section σ of S(φ) defined on W open in Λ, for each integer k ≥ 2, for each ν ∈ W, there exist ν-isolating neighborhoods N_{ν} and N∗_{ν} of S_{σ}(ν), and an open neighborhood V_{ν} of ν, V_{ν} ⊂ W ∩ Λ(N_{ν}) ∩ Λ(N∗_{ν}), so that for each λ ∈ V_{ν}, for j = 1, …, k, there are index pairs ⟨P_{1 j}(λ), P_{0 j}(λ)⟩ relative to N_{ν} and ⟨P ∗_{1 j}(λ), P ∗_{0 j}(λ)⟩ relative to N∗_{ν} for S_{σ}(λ), the former with respect to the forward time flow, the latter with respect to the reverse time flow, satisfying the following statements:

(1) ((P_{1 j}(λ), ∩P_{0 j}(λ)), (P ∗_{1 j}(λ), ∩P ∗_{0 j}(λ))) is AIP(thick) for S_{σ}(λ);

(2) P_{i j−1}(λ) ⊂ int_{N_{ν}} (P_{i j}(ν)) and P ∗_{i j−1}(λ) ⊂ int_{N∗_{ν}}(P ∗_{i j}(ν)) (j = 2, …, k);

(3) P_{i j−1}(ν) ⊂ int_{N_{ν}} (P_{i j}(λ)) and P ∗_{i j−1}(ν) ⊂ int_{N∗_{ν}}(P ∗_{i j}(λ)) (j = 2, …, k);

(4) K ⊂ V_{ν} and K is compact and contractible implies

(i) S_{K} has as index pairs ⟨P̄_{1 j}(K), P̄_{0 j}(K)⟩ and ⟨P̄ ∗_{1 j}(K), P̄ ∗_{0 j}(K)⟩, the former with respect to the forward time flow, the latter with respect to the reverse time flow, where P̄_{i j}(K) := ⋃_{λ∈K} P_{i j}(λ) × {λ} and

P̄ ∗_{i j}(K) := ⋃_{λ∈K} P ∗_{i j}(λ) × {λ} (i = 0, 1; j = 1, …, k);

(ii) λ ∈ K implies that for j = 1, …, k, the map u ↦ (u, λ) induces homotopy equivalences

ῑ_{λ K j}: P_{1 j}(λ)/P_{0 j}(λ) → P̄_{1 j}(K)/P̄_{0 j}(K) ,

and

j̄_{λ K j}: P ∗_{1 j}(λ)/P ∗_{0 j}(λ) → P̄ ∗_{1 j}(K)/P̄ ∗_{0 j}(K) .

5.10 Proof of Theorem 3.2. By Lemma 5.9, there is a cover of [0, 1] by open sets

V_{ν} ⊂ W, ν ∈ W, and index pairs ⟨P_{1 j}(λ), P_{0 j}(λ)⟩ and ⟨P ∗_{1 j}(λ), P ∗_{0 j}(λ)⟩ (j = 1, 2)

satisfying the properties listed in that result. Choose a partition of [0, 1], P : 0 = λ_{0} < … < λ_{n} = 1, subordinate to the cover {V_{ν} : ν ∈ W }, and define K_{j} and S_{j}, j = 1, …, n, relative to this partition as in part (A) of Definition 5.7.

For i = j − 1, j, for j = 1, …, n, let

F_{i j}: C (S_{σ}(λ_{i})) → C (S_{j})

and

G_{i j}: C∗(S_{σ}(λ_{i})) → C∗(S_{j})

be the equivalences of Conley indices induced respectively by the homotopy equivalences

ῑ_{λ_{i} K_{j} 1}: P_{1 1}(λ_{i})/P_{0 1}(λ_{i}) → P̄_{1 1}(K_{j})/P̄_{0 1}(K_{j})

and

j̄_{λ_{i} K_{j} 1}: P ∗_{1 1}(λ_{i})/P ∗_{0 1}(λ_{i}) → P̄ ∗_{1 1}(K_{j})/P̄ ∗_{0 1}(K_{j}) .

It follows that

C (β) = (F_{n n}^{−1} ^{∘} F_{n−1 n})^{∘} ⋯^{∘} (F_{1 1}^{−1} ^{∘} F_{0 1} )

and

C∗(β) = (G_{n n}^{−1} ^{∘} G_{n−1 n})^{∘} ⋯^{∘} (G_{1 1}^{−1} ^{∘} G_{0 1} ) .

For j = 0, …, n, let α_{j} and γ_{j} be defined as in the proof of Theorem 3.3. Again it suffices to show that (5.7) holds.

Claim F_{j j}^{−1}^{∘}F_{j−1 j} and G_{j j}^{−1}^{∘}G_{j−1 j} are respectively determined by the inclusion induced maps on quotients

f̄_{j}: \frac{P_{1 1}(λ_{j−1})}{P_{0 1}(λ_{j−1})} → \frac{P_{1 2}(λ_{j})}{P_{0 2}(λ_{j})}  and  ḡ_{j}: \frac{P ∗_{1 1}(λ_{j−1})}{P ∗_{0 1}(λ_{j−1})} → \frac{P ∗_{1 2}(λ_{j})}{P ∗_{0 2}(λ_{j})}

where the inducing inclusions are respectively

f_{j}: (P_{1 1}(λ_{j−1}), ∩P_{0 1}(λ_{j−1})) ↪ (P_{1 2}(λ_{j}), ∩P_{0 2}(λ_{j}))

and

g_{j}: (P ∗_{1 1}(λ_{j−1}), ∩P ∗_{0 1}(λ_{j−1})) ↪ (P ∗_{1 2}(λ_{j}), ∩P ∗_{0 2}(λ_{j})) .

Furthermore, for i = 0, 1, j = 1, 2, and k = j − 1, j, setting P_{i j k} := P_{i j}(λ_{k}) and P ∗_{i j k} := P ∗_{i j}(λ_{k}) the diagram (5.10)

H(P_{1 2 j}, ∩P_{0 2 j}) ⊗ H(P ∗_{1 2 j}, ∩P ∗_{0 2 j})  ⟶^{𝔏_{P_{2}}}  Ȟ(P_{1 2 j} ∩ P ∗_{1 2 j})
f_{j ∗} ⊗ g_{j ∗}↑  ↑^{}^{ǐ_{P_{2}}^{P_{1}}}
H(P_{1 1 j−1}, ∩P_{0 1 j−1}) ⊗ H(P ∗_{1 1 j−1}, ∩P ∗_{0 1 j−1}) ⟶^{𝔏_{P_{1}}} Ȟ(P_{1 1 j−1} ∩ P ∗_{1 1 j−1})

commutes where ǐ_{P_{2}}^{P_{1}} is the inclusion induced homomorphism as follows immediately from Lemma 2.10.

Assuming the claim that f̄_{j} and ḡ_{j} respectively induce F_{j j}^{−1} ^{∘} F_{j−1 j} and G_{j j}^{−1} ^{∘} G_{j−1 j}, choose

α ∈ H_{m−k}(P_{1 1}(λ_{j−1}), ∩P_{0 1}(λ_{j−1})) representing α_{j−1} ,

and

γ ∈ H_{k}(P ∗_{1 1}(λ_{j−1}), ∩P ∗_{0 1}(λ_{j−1}))  representing γ_{j−1} .

Also choose V ∈ M_{Δ̸ o}^{(2)}(P_{2}) and note that also V ∈ M_{Δ̸ o}^{(2)}(P_{1}). Then

f_{j ∗}α ∈ H_{m−k}(P_{1 2}(λ_{j}), ∩P_{0 2}(λ_{j})) and represents α_{j} ,

and

g_{j ∗}γ ∈ H_{k}(P ∗_{1 2}(λ_{j}), ∩P ∗_{0 2}(λ_{j}))  and represents γ_{j} .

Also, by definition of 𝔏_{V}^{P_{k}} (k = 1, 2) and the commutativity of diagram (5.10)

(5.11) (α_{j−1} ⌢_{•} γ_{j−1})_{V_{1}∩V ∗_{1}} = 𝔏_{V}^{P_{1}}(α ⊗ γ) = 𝔏_{V}^{P_{2}}(f_{j ∗}α ⊗ g_{j ∗}γ) = (α_{j} ⌢_{•} γ_{j})_{V_{1}∩V ∗_{1}}

where the two leftmost (rightmost) intersection pairings are for the isolated invariant set S_{j−1} (respectively, S_{j}) of the flow φ_{λ_{j−1}} (respectively, φ_{λ_{j}}). Combined with our observation (5.8), the string of equalities (5.11) yields

⟨1, (α_{j−1} ⌢_{•} γ_{j−1})_{M}⟩ = ⟨1, (α_{j} ⌢_{•} γ_{j})_{M}⟩ ,

whence (5.7) follows immediately.

It remains to prove the claim. Accordingly, consider the diagram of maps

\frac{P_{1 1}(λ_{j−1})}{P_{0 1}(λ_{j−1})} ⟶^{f̄_{j}} \frac{P_{1 2}(λ_{j})}{P_{0 2}(λ_{j})}
ῑ_{λ_{j−1} K_{j} 1}↓  ↓ῑ_{λ_{j} K_{j} 2}
\frac{P̄_{1 1}(K_{j})}{P̄_{0 1}(K_{j})} ⟶^{φ̄_{j}} \frac{P̄_{1 2}(K_{j})}{P̄_{0 2}(K_{j})}

where φ̄_{j} is induced by the functorial inclusion

φ_{j}: (P̄_{1 1}(K_{j}), ∩P̄_{0 1}(K_{j})) ↪ (P̄_{1 2}(K_{j}), ∩P̄_{0 2}(K_{j}))

and therefore via [K5, Proposition 3.1] has homotopy class a morphism in C (S_{j}).

This diagram is homotopy commutative. To see this, define a homotopy H from φ̄_{j} ^{∘} ῑ_{λ_{j−1} K_{j} 1} to ῑ_{λ_{j} K_{j} 2} ^{∘} f̄_{j} via the formula

H([u], s) := H_{s}([u]) := [(u, (1 − s)λ_{j−1} + sλ_{j})] ∈ \frac{P̄_{1 2}(K_{j})}{P̄_{0 2}(K_{j})}

for [u] ∈ P_{1 1}(λ_{j−1})/P_{0 1}(λ_{j−1}) and s ∈ [0, 1]. Note that H_{s} is well-defined for each s ∈ [0, 1] since it factors through the quotient space (P_{1 2}(ν_{j}) × K_{j})/(P_{0 2}(ν_{j}) × K_{j}) via the maps on quotients induced by the functorial inclusions

(P̄_{1 1}(K_{j}), ∩P̄_{0 1}(K_{j})) ⊂ (P_{1 2}(ν_{j}) × K_{j}, P_{0 2}(ν_{j}) ∩ P_{1 2}(ν_{j}) × K_{j})
⊂ (P̄_{1 2}(K_{j}), ∩P̄_{0 2}(K_{j}))

where ν_{j} ∈ W is chosen so that K_{j} ⊂ V_{ν_{j}}.

It follows that

f̄_{j} ∼ ῑ_{λ_{j} K_{j} 2}^{−1} ^{∘} φ̄_{j} ^{∘} ῑ_{λ_{j−1} K_{j} 1}

where ῑ_{λ_{j} K_{j} 2}^{−1} denotes any homotopy inverse of ῑ_{λ_{j} K_{j} 2}. Then as the homotopy class of φ̄_{j} is a morphism in C (S_{j}), by definition of F_{j−1 j} and the definition of a map of connected simple systems, it follows that the homotopy class of φ̄_{j} ^{∘} ῑ_{λ_{j−1} K_{j} 1} determines F_{j−1 j}: C (S_{σ}(λ_{j−1})) → C (S_{j}). Thus, as ῑ_{λ_{j} K_{j} 2} determines F_{j j}: C (S_{σ}(λ_{j})) → C (S_{j}), necessarily the homotopy class of ῑ_{λ_{j} K_{j} 2}^{−1} determines F_{j j}^{−1}, whence the homotopy class of ῑ_{λ_{j} K_{j} 2}^{−1} ^{∘} φ̄_{j} ^{∘} ῑ_{λ_{j−1} K_{j} 1}, hence of f̄_{j}, determines F_{j j}^{−1} ^{∘} F_{j−1 j} as claimed. Similarly, the homotopy class of ḡ_{j} determines G_{j j}^{−1} ^{∘} G_{j−1 j}. □

5.11 Proof of Theorem 3.1. Let β: [0, 1] → S(φ) be an arbitrary continuous path, let Λ′ = [0, 1], and let the family of flows λ_{β}^{∗}φ and the path β′ that is a section of π_{λ_{β}^{∗}φ}: S(λ_{β}^{∗}φ) → Λ′ be defined using β as in part (B) of Definition 5.7. By that definition, C (β) := C (β′); thus application of Theorem 3.2 to the section β′ immediately yields the desired result. □

It remains to prove Lemma 5.9. The following result is needed first.

5.12 Lemma. Suppose S is an isolated invariant set of a continuous flow on a locally compact Hausdorff space X, (N∗_{1}, N∗_{0}) is a nested index pair for S with respect to the reverse time flow with N∗_{1} isolating S, O is open in X, and N∗_{1} ⊂ O. Then there exists an isolating neighborhood N of S so that N ⊂ O and (N∗_{1}, N∗_{0}) is an index pair relative to N, and there exist index pairs ⟨N_{1}, N_{0}⟩ and ⟨K_{1}, K_{0}⟩ for S relative to N with respect to the forward time flow satisfying

(1) N∗_{1} ∩ N_{0} = ∅ = N_{1} ∩ N∗_{0},

(2) K_{i} ⊂ int_{N}(N_{i}), i = 0, 1,

(3) cl_{X} (N_{1} \ K_{0}) ⊂ int_{X}(N).

5.12.1 Definition. Call ⟨K_{1}, K_{0}⟩ a perturbable index pair if it is an index pair relative to some isolating neighborhood N and if there exists an index pair ⟨N_{1}, N_{0}⟩ relative to N so that conditions (2) and (3) above are satisfied. Note that it is generally impossible for a nested index pair to be perturbable because of conditions (2) and (3). Since it is the index pair ⟨K_{1}, K_{0}⟩ that is being perturbed, this definition of perturbable makes more sense than the definition of perturbable index pair in [K4] where ⟨N_{1}, N_{0}⟩ was called perturbable if (2) and (3) were satisfied for some index pair ⟨K_{1}, K_{0}⟩.

5.12.2 Remark. Lemma 5.12 yields another proof that there exists a pair of index pairs that is AIP(thick) for S. Choose any nested index pair (M∗_{1}, M∗_{0}) for S with respect to the reverse time flow. Then choose s > 0 so that M∗_{0} ⊂ int_{M∗_{1}}(M∗_{1}∩M∗_{0}^{−s}). Set N∗_{1} := M∗_{1} and N∗_{0} := M∗_{1}∩M∗_{0}^{−s} and apply Lemma 5.12. Then the pair ((N_{1}, ∩N_{0}), (N∗_{1}, N∗_{0})) is AIP for S because both index pairs have thick exit set.

Proof of Lemma 5.12. For M any isolating neighborhood, the notations A^{+}(M) and A^{−}(M) denote respectively the forward and backward asymptotic sets of M with respect to the forward time flow; i.e., u ∈ A^{±}(M), if, and only if, u · R^{±} ⊂ M. Similarly, A^{+∗}(M) and A^{−∗}(M) denote respectively the forward and backward asymptotic sets of M with respect to the reverse time flow. Thus, A^{±∗}(M) = A^{∓}(M). With this last equality in mind, interpreting the defining properties of the reverse time index pair (N∗_{1}, N∗_{0}) in terms of the forward time flow as will be done throughout the proof, it follows that N∗_{0} and A^{−}(N∗_{1}) are disjoint closed sets. Thus, choose disjoint open sets V_{1} ⊃ A^{−}(N∗_{1}) and V_{0} ⊃ N∗_{0} with V_{i} ⊂ O (i = 0, 1),

and set F_{0} := ∂_{X}(N∗_{1}) ∩ A^{−}(N∗_{1}).

For each u ∈ F_{0}, there exists t > 0 so that u · t ∈ X \ N∗_{1}. Then compactness of F_{0} and continuity of the flow, imply that there exist strictly positive times t_{1}, …, t_{p} and a finite cover of F_{0} by compact neighborhoods B_{1}, …, B_{p}, each B_{i} a subset of V_{1}, so that B_{i}·t_{i} lies in X \N∗_{1}. Let F′_{0} be the union of the B_{i}·t_{i}, i = 1, …, p. Then F′_{0} is compact and disjoint from F_{0}. Then we can choose a compact neighborhood L_{0} of F_{0} that is disjoint from F′_{0}, but is a subset of the union of the B_{i}, i = 1, …, p. Thus if u ∈ L_{0}, for some i, u · t_{i} lies in F′_{0}, hence does not lie in N∗_{1} ∪ L_{0}. It follows that N∗_{1}∪L_{0} is an isolating neighborhood of S contained in O. Also N∗_{1} is negatively invariant relative to N∗_{1} ∪ L_{0} because any negative orbit segment originating in N∗_{1} and contained in N∗_{1} ∪ L_{0}, but not in N∗_{1}, must exit N∗_{1} through N∗_{0}; hence there must be points on the orbit segment in V_{0} ∩ L_{0} which is impossible. It follows that (N∗_{1}, N∗_{0}) is an index pair relative to N∗_{1} ∪ L_{0}.

Set F_{1} := ∂_{X}(N∗_{1} ∪ L_{0}) ∩ A^{−}(N∗_{1} ∪ L_{0}), and note F_{1} is disjoint from N∗_{1}. For as L_{0} is a neighborhood of F_{0}, certainly F_{1} ∩ F_{0} = ∅, whence if u ∈ N∗_{1} ∩ F_{1}, then u ∉ A^{−}(N∗_{1}). Thus, where τ∗_{N∗_{1}} is the entrance time map of N∗_{1} relative to the forward time flow, i.e., the exit time map relative to the reverse time flow,

u′ := u · −τ∗_{N∗_{1}}(u) ∈ N∗_{0} ⊂ V_{0}

and there exists ε > 0 so that u′ · [−ε, 0] ⊂ V_{0}. Also, by definition of u′, u′ · [−ε, 0] intersects V_{0} \ N∗_{1}. However, as u′ ∈ A^{−}(N∗_{1} ∪ L_{0}), it follows that V_{0} intersects L_{0}, an impossibility.

Thus by an argument similar to that used to choose L_{0}, we can choose a compact neighborhood L_{1} of F_{1} with the properties: (i) L_{1} ⊂ O and is disjoint from N∗_{1} ∪ V_{0} ∪ F′_{0}, (ii) there exists a compact neighborhood F′_{1}, disjoint from N∗_{1} ∪ L_{0} ∪ L_{1}, and strictly positive times s_{1}, …, s_{q} so that for each u ∈ L_{1}, for some i, u·s_{i} ∈ F′_{1}.

Define N by

N := N∗_{1} ∪ L_{0} ∪ L_{1} .

Then from the properties of L_{1}, it follows, as before, that N is an isolating neighborhood for S and is a subset of O, and also (N∗_{1}, N∗_{0}) is an index pair relative to N for the reverse time flow.

We note that N∗_{1}, hence also N \ L_{1}, is an N-neighborhood of A^{+}(N). For if u ∈ L_{0} ∪ L_{1}, then for some t > 0, u · t ∈ F′_{0} ∪ F′_{1}, which is disjoint from N. Thus

A^{+}(N) ⊂ N \ (L_{0} ∪ L_{1}) ⊂ N∗_{1} ,

which gives the desired result. Thus L_{1} is disjoint from A^{+}(N). Thus, recalling that for general K ⊂ N, P′(K, N) (respectively, P (K, N)) denotes the smallest (respectively, smallest compact) positively invariant relative to N subset of N containing K, defining K_{0} := P (L_{1}, N), we get via [K4, Proposition 4.1] that

L_{1} ⊂ K_{0} = P′(L_{1}, N) .

Also, K_{0} is disjoint from N∗_{1}. For suppose u ∈ K_{0} ∩ N∗_{1}. Then for some u′ ∈ L_{1} and t′ > 0,

u = u′ · t′ and u′ · [0, t′] ⊂ N .

Set t′′ := sup{t > 0 : u′ · [0, t] ⊂ N \ N∗_{1}}. Because L_{1} is disjoint from N∗_{1}, necessarily 0 < t′′ ≤ t′. Set u′′ = u′ · t′′, and note u′′ ∈ N∗_{1}. By definition of t′′ it is immediate that u′′ ∉ A^{−}(N∗_{1}), whence by the exit property of index pairs, necessarily u′′ ∈ N∗_{0}. However, as cl_{X} (N \ N∗_{1}) ⊂ L_{0} ∪L_{1}, also u′′ ∈ L_{0} ∪L_{1}. This is impossible since N∗_{0} and L_{0} ∪ L_{1} are disjoint.

As K_{0} is disjoint from N∗_{1}, choose Z_{0}, a closed N-neighborhood of K_{0}, disjoint from N∗_{1}. Set N_{0} := P (Z_{0}, N). Then N_{0} is an N-neighborhood of K_{0}, and by an argument analogous to the one just given, N_{0} is disjoint from N∗_{1}.

As A^{−}(N) is disjoint from N∗_{0} and by the construction of L_{0} and L_{1} as neighborhoods of F_{0} and F_{1} respectively, we can choose an open neighborhood W of A^{−}(N) disjoint from a neighborhood of N∗_{0} and which has the additional properties that cl_{X} (W ∩ N) is compact and

∂_{X}(N) ∩ cl_{X} (W ∩ N) ⊂ N \ (N∗_{1} ∪ L_{0}) = L_{1} \ L_{0} .

Now by [K4, Proposition 4.1(5)], there exists Z′ a compact N-neighborhood of A^{−}(N) so that A^{−}(N) ⊂ Z ⊂ Z′ implies P (Z, N) ⊂ W. Thus choose Z_{0} to be a compact N-neighborhood of A^{−}(N) and an X-neighborhood of S satisfying Z_{0} ⊂ Z′. Set N_{1} := P (Z_{0}, N). Then N_{1} isolates S, is positively invariant relative to N, and is a subset of W ∩ N. Also, N_{1} is an N-neighborhood of A^{−}(N) since it contains Z_{0}; hence, for some V an open subset of W,

A^{−}(N) ⊂ V ∩ N ⊂ int_{N}(N_{1}) .

Applying an analogous argument with V replacing W, allows us to find K_{1} ⊂ V ∩N such that K_{1} isolates S and is positively invariant relative to N . By definition of V, it is immediate that K_{1} lies in the interior of N_{1} relative to N.

It follows that ⟨N_{1}, N_{0}⟩ and ⟨K_{1}, K_{0}⟩ are index pairs for S with the desired properties. The only property which is not obviously satisfied is the exit property. However, if u ∈ N_{1} \ A^{+}(N), and also u ∉ K_{0}, set

u′ := u · τ_{N}(u) .

Note u′ ∈ ∂_{X}(N) ∩ cl_{X} (W ∩ N), whence by our choice of W,

u′ ∈ L_{1} \ L_{0} ⊂ K_{0} .

Since N_{1} is positively invariant relative to N and since K_{0} ⊂ N_{0} and K_{1} ⊂ N_{1}, it follows immediately that the exit property is satisfied for both pairs. □

5.13 Proof of Lemma 5.9. Choose an isolating neighborhood N∗_{ν} of S_{σ}(ν). By the construction of index pairs given in [C] and by [K4, Lemma 4.9] (see Remark 3 following Theorem 4.2 of [K4]) we can find, for j = 0, …, k+1, index pairs ⟨N∗_{1 j}, N∗_{0 j}⟩ for S_{σ}(ν) relative to N∗_{ν} and the reverse time flow satisfying:

(1) N∗_{i j} ⊂ int_{N∗_{ν}}(N∗_{i j+1}) for i = 0, 1, for j = 0, …, k + 1;

(2) cl (N∗_{1 j+1} \ N∗_{0 j}) ⊂ int_{X}(N∗_{ν}) for j = 0, …, k.

Then by Lemma 5.12 we can find an isolating neighborhood N_{ν} of S_{σ}(ν) and index pairs ⟨N_{1}, N_{0}⟩ and ⟨K_{1}, K_{0}⟩ for S_{σ}(ν) relative to N_{ν} and the forward time flow satisfying properties (2) and (3) in the statement of that lemma and also

(5.12)  N_{1} ∩ N∗_{0 k+1} ∩ N∗_{1 k+1} = ∅ = N_{0} ∩ N∗_{1 k+1} .

Then setting N_{i k+1} := N_{i} and N_{i 0} := K_{i} (i = 0, 1), by [K4, Lemma 4.9] we can find additional sets N_{i j} (i = 0, 1, j = 1, …, k) so that

(3) N_{i j} ⊂ int_{N_{ν}} (N_{i j+1}) for i = 0, 1, for j = 0, …, k;

(4) cl (N_{1 j+1} \ N_{0 j}) ⊂ int_{X}(N_{ν}) for j = 0, …, k.

(5) ⟨N_{1 j}, N_{0 j}⟩ is an index pair for S_{σ}(ν) relative to N_{ν} and the forward time

flow for j = 0, …, k + 1. Note that as a consequence of (5.12) and the inclusion relations in items (1) and (3), the pair of index pairs

((N_{1 j}, ∩N_{0 j}), (N∗_{1 j}, ∩N∗_{0 j})) j = 1, …, k + 1

is AIP(thick) for S_{σ}(ν).

For every compact pair (N, K) in X and λ ∈ W define P (K, N : λ) to be the intersection of all the subsets of N that contain K and are compact and positively invariant relative to N relative to the flow φ_{λ}. The set P ∗(K, N : λ) is defined analogously with respect to the reverse time flow associated to φ_{λ}. Then for λ ∈ W, for i = 0, 1, for j = 0, …, k, define

P_{i j}(λ) := P (N_{i j}, N_{ν} : λ) and P ∗_{i j}(λ) := P ∗(N∗_{i j}, N∗_{ν} : λ) .

Note that P_{i j}(ν) = N_{i j} and P ∗_{i j}(ν) = N∗_{i j}.

By the Perturbation Theorem for Index Pairs, [K4,Theorem 4.2] also see [C, §IV1.2A], we get that there exists V_{ν} open, ν ∈ V_{ν} ⊂ W ∩ Λ(N_{ν}) ∩ Λ(N∗_{ν}) so that for λ ∈ V_{ν},

(6) for j = 0, …, k, ⟨P_{1 j}(λ), P_{0 j}(λ)⟩ and ⟨P ∗_{1 j}(λ), P ∗_{0 j}(λ)⟩ are index pairs for S_{σ}(λ), the former relative to N_{ν} and the flow φ_{λ}, the latter relative to N∗_{ν} and the reverse time flow associated to φ_{λ};

(7) N_{i j} ⊂ P_{i j}(λ) ⊂ int_{N_{ν}} (N_{i j+1}) for i = 0, 1, for j = 0, …, k;

(8) N∗_{i j} ⊂ P ∗_{i j}(λ) ⊂ int_{N∗_{ν}}(N∗_{i j+1}) for i = 0, 1, for j = 0, …, k.

By (5.12) and the inclusion relations in (1), (3), (7) and (8), it follows that for λ ∈ V_{ν}, the pair of index pairs

((P_{1 j}(λ), ∩P_{0 j}(λ)), (P ∗_{1 j}(λ), ∩P ∗_{0 j}(λ))) (j = 1, …, k)

is AIP(thick) for S_{σ}(λ).

For i, j fixed, the sets P_{i j}(λ) and P ∗_{i j}(λ) vary upper semi-continuously with λ, so that for K ⊂ V_{ν} compact, the sets

P̄_{i j}(K) := ⋃_{λ∈K} P_{i j}(λ) × {λ} and P̄ ∗_{i j}(K) := ⋃_{λ∈K} P_{i j}(λ) × {λ}

are compact (see [K4, Lemma 4.8]). It follows easily that

⟨P̄_{1 j}(K), P̄_{0 j}(K)⟩ and ⟨P̄ ∗_{1 j}(K), P̄ ∗_{0 j}(K)⟩ (j = 1, …, k)

are index pairs for S_{K}, the former with respect to the forward time flow on X × K, the latter with respect to the reverse time flow on X × K.

Further, by Lemma 5.6 we can shrink V_{ν} if necessary so that for λ ∈ V_{ν}, for K ⊂ V_{ν} compact and contractible, the maps on quotients

ῑ_{λ K j}: P_{1 j}(λ)/P_{0 j}(λ) → P̄_{1 j}(K)/P̄_{0 j}(K)

and

j̄_{λ K j}: P ∗_{1 j}(λ)/P ∗_{0 j}(λ) → P̄ ∗_{1 j}(K)/P̄ ∗_{0 j}(K)

are homotopy equivalences since ῑ_{λ K j} and j̄_{λ K j} respectively define the maps of connected simple systems F (λ, K): C (S_{σ}(λ)) → C (S_{K}) and G(λ, K): C∗(S_{σ}(λ)) → C∗(S_{K}) for j = 1, …, k. □

5.14 Proof of Theorem 3.7. Without loss of generality let us assume that β is the restriction of a section σ as in the statement of Theorem 3.2. Also, let us assume that 0 ≤ s < 1 and prove ℓ(β, α, γ) right continuous at s; for 0 < s ≤ 1, left continuity at s ∈ ]0, 1] is proved similarly. The proof of right continuity at s is essentially already contained in the proof of Theorem 3.2 and in particular follows from the string of equalities (5.11) after some minor restrictions are placed on a few of the choices made in the course of the derivation of (5.11). Throughout the proof we use the same notation as in the proof of Theorem 3.2.

To begin, suppose ℓ(β, α, γ)(s) ∈ N (ζ̌, S, λ, N, U) ∈ B_{k}, whence

(1) s = λ_{β}(s) ∈ U and S_{β}(s) has N as λ_{β}(s)-isolating neighborhood;

(2) π_{N}^{S_{β}(s)}(C (β^{s})_{∗} α ⌢_{•} C∗(β^{s})_{∗} γ) = π_{N}^{S} ζ̌.

To prove right continuity at s_{0} it suffices to exhibit δ > 0 so that if 0 < s′ − s ≤ δ, then the analogues of (1) and (2) obtained by replacing s with s′ hold.

By continuity of σ, choose δ_{0} so that s′ ∈ [0, 1] and |s − s′| ≤ δ_{0} implies λ_{σ}(s′) ∈ U and S_{σ}(s′) has N as λ_{β}(s′)-isolating neighborhood, whence σ(λ) = σ_{N}(λ) if |s − λ| ≤ δ_{0}. Apply Lemma 5.9 and choose a partition P as in §5.10, the proof of Theorem 3.2, where without loss of generality we can additionally assume that mesh(P) ≤ δ_{0} and that s is one of the partition points of P, i.e, s = λ_{i−1} where i ∈ {1, …, n}. Observe that the proof of Lemma 5.9 allows us to assume without loss of generality that for each ν ∈ J_{i} := [λ_{i−1}, λ_{i}], the isolating neighborhoods N_{ν} and N∗_{ν} guaranteed by that lemma lie interior to N. It then follows that for each λ ∈ J_{i}, for j = 1, 2, the pair of admissible index pairs P_{j}(λ) has the property P_{1 j}(λ) ∩ P ∗_{1 j}(λ) ⊂ int(N). It follows that when choosing V ∈ M_{Δ̸ o}^{(2)}(P_{2}) as in the proof of Theorem 3.2, it can be assumed that V_{1} ∩ V ∗_{1} ⊂ N.

Set δ := mesh(P) and suppose 0 < s′−λ_{i−1} ≤ δ. Then without loss of generality it can be assumed that s′ = λ_{i}. For if not refine the partition P by adding s′ as a partition point. Then as in the proof of Theorem 3.2 we arrive at the string of equalities (5.11) from which it is immediate that

(5.13)  (α_{i−1} ⌢_{•} γ_{i−1})_{N} = (α_{i} ⌢_{•} γ_{i})_{N}

because V_{1} ∩ V ∗_{1} ⊂ N. The equality (5.13) and (2) together yield that

π_{N}^{S_{σ}(s′)}(C (β^{s′})_{∗} α ⌢_{•} C∗(β^{s′})_{∗} γ) = π_{N}^{S}(ζ̌) .

The choice of δ implies that λ_{β}(s′) ∈ U and S_{β}(s′) has N as λ_{β}(s′)-isolating neighborhood. □