HOMOTOPICAL DYNAMICS III: REAL SINGULARITIES AND HAMILTONIAN FLOWS.

OCTAVIAN CORNEA

Abstract. On the space of non-depraved [8] isolated singularities, we consider the stable equivalence relation induced by smooth deformations whose asymptotic behaviour is controlled by the Palais-Smale condition. It is shown that the resulting space of equivalence classes admits a canonical semi-ring structure and is isomorphic to the semi-ring of stable homotopy classes of CW -complexes.

In an application to Hamiltonian dynamics we relate the existence of bounded and periodic orbits on non-compact level hypersurfaces of Palais-Smale Hamiltonians with just one singularity which is non-depraved to the lack of self-duality (in the sense of Spanier-Whitehead) of the sublink of the singularity.

1. Introduction.

This paper is concerned with the dynamics of the gradient and hamiltonian flows associated to real, isolated, non-depraved (in the sense of Goreski and MacPherson [8]) singularities. This is a fairly general class of singularities that includes locally analytic singularities. Its definition will be recalled in the second section.

1.1. Deformations. We first study an equivalence relation on the set of these singularities. The motivation and origin of this relation lies in nonlinear analysis and the calculus of variations and, in particular, in the notion of continuation in Conley index theory. From our point of view, two nondepraved singular germs f and g are equivalent if the asymptotic geometric behaviour of their gradient flows is (stably and uniformly) cobordant.

One can formalize this as follows. For the two germs f and g consider extensions to R^{n} that do not add any new critical points and have asymptotically ample gradients that define global flows (the ”ampleness” condition is that they satisfy a version of the Palais-Smale condition; it will be shown that such extensions always exist). Such extensions are called PS-extensions. The two germs are stably Palais-Smale equivalent if, there are such extensions which, after adding a quadratic form in independent variables, can be deformed one to the other via a deformation that satisfies a uniform variant of the condition above. Let Sing_{n}^{P S} be the set of equivalence classes of germs of non-depraved singularities f : D^{n} ⟶ R with respect to stable Palais-Smale equivalence (it is a non-trivial fact that this is indeed an equivalence relation).

Date: September 9, 2000 (revised)

2000 Mathematics Subject Classification. Primary 37B30, 57R45; Secondary 37J45.

As only asymptotic restrictions are imposed along the deformation relating f to g, and, therefore, bifurcations and non isolated singular points can occur, it might appear that this equivalence relation is very weak.

The first key idea of this paper is that, in fact, it is precisely as rigid as needed to reduce the study of these singularities to homotopy theory.

To state the first main theorem, let CW_{n}^{S} be the abelian semigroup of stable homotopy classes of CW -complexes that admit a thickening [23] in S^{n−1} (the operation is the wedge). This semigroup supports an obvious involution induced by X → X^{∗} with X^{∗} the complement of X in S^{n−1}. There is also an operation CW_{n}^{S} × CW_{k}^{S} ⟶ CW_{k+n}^{S} given by the join of complexes. This is again compatible with the involution. The equatorial inclusion S^{n−1} ↪ S^{n} allows one to assemble this data in a semiring CW_{∗}^{S}.

Theorem 1.1. On Sing_{n}^{P S} there is a “fusion of singularities” operation such that the application associating to a singularity germ f its sublink A_{f} induces a semi-group isomorphism Φ_{n} : Sing_{n}^{P S} ≈ CW_{n}^{S} compatible with the involution f → −f. Moreover, there is also an “exterior sum operation” ⊕ : Sing_{n}^{P S} × Sing_{k}^{P S} ⟶ Sing_{n+k}^{P S} induced by (f, g) → f + g . There are semigroup inclusions Sing_{n}^{P S} ↪ Sing_{n+1}^{P S} given by [f(x)] → [f(x)] ⊕ [y^{2}] with (x, y) ∈ R^{n} × R. Let Sing_{∗}^{P S} be the resulting filtered semiring. The maps Φ_{∗} induce a semiring isomorphism Sing_{∗}^{P S} ≈ CW_{∗}^{S}.

For a non-depraved germ f and for a sufficiently small δ > 0, the sublink A_{f} of the singularity is the homeomorphism type of the space

f^{−1}(−∞, f(0)] ⋂ S(δ, 0)

and is independent of δ (the argument will be recalled in the second section). The Conley index of the critical point of f is the homotopy type of ΣA_{f}. Therefore, it results from Theorem 1.1 that, if the Conley indexes of two non-depraved critical points are the same, then, after possibly adding some quadratic form, the two singularity germs can be related by continuation. This gives a partial answer to Conley’s question A in §IV.8.1 in [1].

1.2. Application to Hamiltonian dynamics. Our methods apply to C^{2} Hamiltonians \overline{f} : R^{2n} ⟶ R that are Palais-Smale and that have a single singularity which is of the type above.

In other words we consider the Hamiltonian flow associated to a C^{2} PS extension of some non-depraved singular germ f with respect to some symplectic form ω on R^{2n}.

The major question about this flow is whether it has closed orbits on some constant energy hypersuface of the form S_{a} = \overline{f}^{−1}(a). Most known results detecting closed orbits of Hamiltonian flows apply to the case when S_{a} is compact (for a survey see [9]). Compactness of S_{a} corresponds in our context to the very special case of f being a local extremum. We will assume in the following that this is not the case. Then the hypersurface S_{a} is non-compact (for all a).

In this situation it is natural to look first for the existence of bounded orbits.

A second central idea of the paper is that the phenomenon of persistent, infinitely many, bounded orbits is implied by a lack of self-duality.

Here is how to make this idea precise. We will say that a non-depraved germ f has bounded orbits if the following property holds for any symplectic form ω: for any PS extension of f the associated ω- Hamiltonian flow has an orbit inside D(R, 0) which intersects notrivially S(R, 0) (D(R, 0) is the closed euclidean disk of radius R and center 0 and S(R, 0) is its boundary).

We say that a non depraved germ f is self dual if f and −f are stably PS equivalent.

Theorem 1.2. Assume f : D^{2n} ⟶ R is a non-depraved germ. If f is not self-dual, then f has bounded orbits.

I do not know under what precise circumstances the presence of selfduality leads to the non-existence of bounded orbits.

Notice that, by Theorem 1.1, f is self-dual iff A_{f} is self-dual in the sense of Spanier-Whitehead [20] (or, in other words, possibly after some suspensions, A_{f} and the closure of its complement in S^{n−1} have the same homotopy type). Threfore lack of self-duality depends only on the stable PS equivalence class of f. Other consequences of the fundamental relation between flow inversion and Spanier-Whitehead duality have been exploited in the first papers of this series [4][5].

We are also interested in detecting bounded orbits that persist not only under PS deformations but also with respect to stabilization. We will say that the non-depraved germ f has stable bounded orbits if, for any nondegenerate quadratic form q, any non-depraved germ in [f]⊕[q] has bounded orbits.

It is natural to neglect self-dual germs. Consider the following equivalence relation on the space of all singular non-depraved germs. We say that two such germs f and g are tamely equivalent if there are self-dual, non-depraved germs T_{i} and non-degenerate quadratic forms q_{i} for i = 1, 2 such that f + T_{1} + q_{1} is stably Palais-Smale equivalent to g + T_{2} + q_{2} (this is easily seen to be an equivalence relation).

Let Sing^{T} be the resulting set of equivalence classes. Let \overline{CW} be the set of equivalence classes obtained from CW_{∗}^{S} by identifying X and ΣX for each CW -complex X and by inverting all self-dual complexes (ΣX is the suspension of X). It is easy to see that \overline{CW} is a group.

Corollary 1.3. The exterior sum produces a group structure on Sing^{T} and the semiring morphism Φ induces a group isomorphism \overline{Φ} : Sing^{T} ⟶ \overline{CW}. If \overline{Φ}(f) ≠ 0, then f has stable bounded orbits.

Obviously, we are also interested in producing some closed orbits. We will understand here by closed orbit one in the usual sense or one that closes at a stationary point.

Corollary 1.4. Let f : D^{2n} ⟶ R be a non-depraved germ. Every PS extension of f has a Whitney C^{2} neighbourhood containing a dense family of functions each of whose induced Hamiltonian flow has at least rk(H_{∗}(A_{f}; Z))− rk(H_{n−1}(A_{f}; Z)) non-constant closed orbits.

1.3. Structure of the paper. The proofs of Theorems 1.1 and 1.2 are based on elements of singularity theory and some basic facts concerning the Conley index [1][19]. That these two techniques have something in common has been suggested before, for example in [8] [2].

The steps leading to these proofs have some interest in themselves. They are presented as follows.

Section 2 contains, besides basic recalls of properties of non-depraved singularities and the Palais-Smale condition, one key fact relating these two notions: any non-depraved germ admits a PS extension.

The purpose of the third section is to prove that if two non-depraved germs f, g are related by a deformation satisfying a Palais-Smale type condition, then ΣA_{f} ≃ ΣA_{g} (where ≃ is homotopy equivalence). Conversely, if two germs have sub links of the same stable homotopy type, then their stabilizations can be related by a PS deformation. For the proof of the first implication we make use of the continuation properties of the Conley index. As a corollary we introduce stable PS equivalence.

In §4 we describe the operations in Sing_{∗}^{S} and prove Theorem 1.1. We then discuss the application to Hamiltonian dynamics and prove Theorem 1.2 and its corollaries. This makes again use of the properties of the Conley index. The closed orbit statement is based on the C^{1} closing lemma of Pugh and Robinson [15].

To increase the accessibility of the paper we have tried to make it as self-contained as possible. We review and indicate the proofs of the needed elements of singularity theory. We also recall the basic definitions and properties relative to the Conley index.

2. All non-depraved singularities have Palais-Smale extensions.

2.1. Non-depraved singularities. We denote by || || the standard euclidean norm on R^{n}. Let D^{n} = D(δ, 0) = {y : ||y|| ≤ δ}, S(δ, 0) = ∂D(δ, 0). We will use below the term “smooth” to mean C^{r} with r ≥ 2.

Following [8] we recall the definition of non-depraved singularities.

Definition 1. Let f : (D^{n}, 0) ⟶ (R, 0) be a smooth function. The point 0 is a non-depraved singularity of f if it is an isolated singularity and for any sequence x_{n} such that lim_{n→∞} x_{n} = 0, lim_{n→∞}(∇f(x_{n})/||∇f(x_{n})||) = v ∈ S(1, 0), lim_{n→∞} x_{n}/||x_{n}|| = w we either have v • w = 0 or sign((v • w)f(x_{n})) = 1 for all sufficiently big n (here • is the scalar product of vectors in R^{n}; sign(a) is the sign of a and is taken to be 0 if a = 0).

Obviously, one could make the same definition for a metric different from the standard one. However, the non-depravity conditon is independent of the metric used on R^{n} and is invariant with respect to diffeomorphisms leaving invariant the origin [8]. If 0 is a non-depraved singularity of f the following condition is also satisfied:

∗ there exists δ > 0 such that for all ϵ < δ the singular hypersurface f^{−1}(0) intersects transversely the sphere S(ϵ, 0)

Indeed, if ∗ would not be satisfied there would be a sequence x_{n} with lim_{n→+∞} x_{n} = 0 and f(x_{n}) = 0, ∇f(x_{n}) = λ_{n}x_{n}. We may assume that x_{n}/||x_{n}|| and ∇f(x_{n})/||∇f(x_{n})|| both converge. But this immediately leads to a contradiction with the non-depravity condition.

In the following we will say that the isolated singularity 0 of a smooth function f : (D^{n}, 0) ⟶ (R, 0) is neat if it satisfies the condition ∗ above (in particular, non-depraved implies neat).

The basic fact that we need to recall is:

Lemma 2.1. If 0 is a neat singularity of f, then for some small ϵ > 0 the homeomorphism type of the sublink of f, f^{−1}(−∞, 0] ⋂ S(δ, 0), is independent of δ < ϵ. This homeomorphism type will be denoted by A_{f}(0) (or A_{f}).

The argument for this is classical [13]. Indeed, let H = f^{−1}(0). Assume that for δ ≤ ϵ the intersection of H with S(δ, 0) is transversal. Let H′ = H ⋂ D(ϵ, 0). Let V be the vector field on D(ϵ, 0) given by the projection of the gradient vector field X of the function x ⟶ ||x||^{2} in the direction of −∇f/||∇f||. Similarly, let W be obtained by projecting X on the hyperplane orthogonal to ∇f. Let ν : D(ϵ, 0) ⟶ [0, 1] be a smooth function such that ν^{−1}(0) = H′, ν^{−1}(1) is a neighbourhood of f^{−1}((−∞, −τ] ⋃[τ, ∞)) ⋂ S(ϵ, 0) with τ so small that the intersection of S(ϵ, 0) with f^{−1}(τ′) is transverse for all τ′ ∈ [−τ, τ] (such a ν may be constructed by a suitable modification of f^{2}). Consider the vector field U = W + νV. This is smooth except at 0, is tangent to H′ and points out on S(ϵ′, 0) for all 0 < ϵ′ ≤ ϵ. It immediately follows that the flow induced by U provides the homeomorphisms between f^{−1}(−∞, 0] ⋂ S(δ, 0) and f^{−1}(−∞, 0] ⋂ S(ϵ, 0) for all 0 < δ ≤ ϵ.

Remark 1. a. A main problem with neat singularities is that the neatness condition, in contrast to the non-depravity one, depends strongly on the metric chosen on R^{n}.

b. A non-depraved singularity f also has the property that the pair (f^{−1}(0) − {0}, 0) forms a Whitney stratification [24][22]. This last condition is again independent of the metric, is invariant under diffeomorphisms and it implies neatness. A singularity satisfying only this stratification condition was called reasonable in [2].

c. Given a neat singularity germ f consider the link of f, f^{−1}(0) ⋂ S(δ, 0) (this is, of course, the boundary of the sublink). The same arguments as above show that the homeomorphism type of the link is also invariant with respect to (sufficiently small) δ. We will denote the link of f by L_{f}.

The following local structure result will be useful. Assume f : R^{n} ⟶ R is a smooth function with a single critical point at the origin and f(0) = 0. Consider the flow γ induced by −∇f, the negative of the gradient of f. Consider the cylindrical neighbourhoods of 0 defined by U(ϵ, δ) = f^{−1}[−ϵ, ϵ] ⋂{x ∈ R^{n} : ∃t ∈ R ⋃{∞, −∞}, f(γ_{t}x) = 0, ||γ_{t}x|| ≤ δ}. Let A(ϵ, δ) = U(ϵ, δ) ⋂ f^{−1}(−ϵ).

Lemma 2.2. If 0 is a non-depraved singularity and δ, ϵ are small enough, then there is a homeomorphism of pairs:

(U(ϵ, δ), A(ϵ, δ)) ≈ (D^{n}, A_{f}(0))

Proof. It is clear that the different small neighbourhoods U(ϵ, δ) are homeomorphic. This is true even for neat singularities due to the conical structure of f^{−1}(0) ⋂ D(τ, 0) for small τ (this is an immediate consequence of Lemma 2.1). For this reason we take ϵ, δ and δ′ small enough such that U(ϵ, δ) ⊂ D(δ′, 0) and f^{−1}(0) intersects transversely all spheres of radius smaller than δ′. As above construct a vector field U smooth (except at 0) in D(δ′, 0) that points out on the boundary and is tangent to f^{−1}(0) − {0}. Let V be the smooth vector field which is the component of U orthogonal to ∇f. Consider the flow associated to V. If we take ϵ small enough this flow will cary U(δ, ϵ) into D(δ′, 0) ⋂ f^{−1}([−ϵ, ϵ]) providing a homeomorphism between these two sets which restricts to a homeomorphism A(δ, ϵ) ≈ D(δ′, 0) ⋂ f^{−1}(−ϵ). On the other hand, another modification of the vector field U provides a homeomorphism of pairs (D(δ′, 0) ⋂ f^{−1}([−ϵ, ϵ]), D(δ′, 0) ⋂ f^{−1}(−ϵ)) ≈ (D(δ′, 0), S(δ′, 0) ⋂ f^{−1}(−∞, 0]). More precisely, one adds to U a vector field that is null outside a neighbourhood of |f|^{−1}([ϵ′, ϵ]) where it points in the direction of sign(f)∇f. Let X′′ be the resulting vector field. Non-depravity implies that, for sufficiently small ϵ, and for all x satisfying ||x|| ≤ ϵ, ∇f(x) = λx, λ ∈ R, we have sign(λ) = sign(f). Because of this, when ϵ′ < ϵ are sufficiently small, we see that the vector field, X′′, will point “out” on the boundary of D(δ′, 0) ⋂ f^{−1}([−ϵ, ϵ]) whithout adding any new stationary points. The induced flow provides the homeomorphism indicated.

Remark 2. a. The cylindrical neighbourhoods U(ϵ, δ) appear to have been first introduced in [18]. Their properties have also been exploited in [6].

b. For singularities that are only neat, a simple invertible cobordism argument (as in [2]) shows the existence of a homotopy equivalence of pairs (U(ϵ, δ), A(ϵ, δ)) ≃ (D^{n}, A_{f}(0)).

2.2. Palais-Smale extensions. Let M be a smooth manifold and let α be a riemannian metric on M. Let f : M ⟶ R be a smooth function. We denote by ∇^{α}f the α gradient of f. Recall that the function f satisfies the Palais-Smale condition (or is PS) with respect to α [14] if for any m ∈ R a sequence x_{n} that verifies ∇^{α}f(x_{n}) → 0 and |f(x_{n})| ≤ m contains a convergent subsequence.

Whenever speaking about a PS function we will assume implicitely that the relevant metric has been already fixed.

Proposition 2.3. For any non-depraved, singular germ, f, there is a disk D^{n}, a riemannian metric α on R^{n} and a function (which will be called a Palais-Smale extension of f) \overline{f} : R^{n} → R with a single critical point and such that ∇^{α}\overline{f} induces a flow, \overline{f}|_{D^{n}} = f and \overline{f} is PS with respect to α.

Proof. Let γ be the (partial) flow induced by −∇f. As above, for ϵ and δ small enough, let U = U(ϵ, δ) be a cylindrical neighbourhood of 0. The interior of U is homeomorphic to R^{n} . Notice that ∇f induces only a partially defined flow on Int(U) (as each flow line is defined only for finite time). We intend to change the canonical metric on U as well as the function f such that this defficiency will be corrected and with respect to the new metric the modified function will satisfy the desired property. Let ϵ′ < ϵ. Notice that W = Int(U(ϵ, δ)) − Int(U(ϵ′, δ)) is diffeomorphic to the disjoint union of A × (0, 1] ∐ B × (0, 1] where A = A × {0} = U ⋂ f^{−1}(ϵ) and B = B × {0} = U ⋂ f^{−1}(−ϵ). We will use for the set W coordinates of the form (y, t) where y ∈ A ∐ B and t gives the direction of the negative of the gradient of f. In these coordinates, let ν be the standard metric and let ∂/∂t be the unit tangent vector in the direction of t. We will focus only on the part, W′, of W diffeomorphic to A × (0, 1]. Let u, v : U ⟶ R be smooth functions such that u(x) = 1 = v(x) for x ∈ U − Int(W′), u and v depend only on t inside W′, u(0) = 0 = v(0), u(1) = 1 = v(1), u′(t), v′(t) > 0 for t ∈ [0, 1), u(t) = tv(t) in a small neighbourhood of 0 and lim_{t→0} tv′(t)/v(t) = 1.

Define a new metric on Int(U) by ν′ = (1/u^{2})ν. Modify the function f by defining f′ = (1/v)f on W′. Clearly, the negative of the gradient of f′ with respect to ν′, −∇′f′, points in the direction t. Its value is given by ν′(∇′f′, ∂/∂t) = ∂f′/∂t. This means

−∇′f′ = ((v′u^{2}/v^{2})f − (u^{2}/v)∂f/∂t)∂/∂t

on W′.

Notice that ∂f/∂t is negative on W′. Hence, the ν′ norm of ∇′f′ is equal to (v′u/v^{2})f − (u/v)∂f/∂t. When t tends to 0 the sum goes to ϵ.

Therefore, this expression is bounded from above on Int(U). Moreover, when x ≠ 0 the sum is strictly positive. Therefore, for some small ω > 0, on Int(U) − D(ω, 0) the ν′ norm of ∇′f′ is bounded both from above and from below. The existence of a bound from above implies that the flow induced by −∇′f′ is well defined for infinite negative time. A similar construction applied to W − W′ leads to the desired metric α and function \overline{f}.

Notice that the metric α coincides with the canonical one on a small disk around the origin. Of course, the same construction could have been performed by starting with any other riemannian metric around the origin (because the non-depravity condition is independent of the metric).

Remark 3. a. At first sight the existence of Palais-Smale extensions might seem an obvious fact. However, it is easy to construct a Morse function on D^{n} with only two critical points, both non-degenerate, and which does not admit a PS extension to R^{n}. Indeed, consider the height function h on the sphere S^{n}. Let S and N be its two critical points and let E ∈ h^{−1}(0). Let p be the stereographic projection from E onto a plane tangent to the antipodal of E. Let D^{n} be the image of a complement of a disk around E. The function h′ = h ∘ p^{−1} : D^{n} ⟶ R does not admit a PS extension to R^{n}.

It remains open what is the exact class of singularities for which the extension result of the proposition remains valid.

b. For any two non-depraved germs f and g there is a smooth function F : R^{n} × [0, 1] ⟶ R with F_{0} and F_{1} Palais-Smale with unique critical points respectively extending f and g. Indeed, one just has to use a smooth homotopy of the two extensions \overline{f} and \overline{g} provided by the proposition as well as a smooth deformation of the metrics.

We will also need some local properties of sums of non-depraved singularities. We recall that the join X ∗ Y of two spaces X, Y is given by CX × Y ⋃_{X×Y} X × CY where CX is the cone on X.

Lemma 2.4. a. If h : D^{n} ⟶ R and g : D^{k} ⟶ R are non-depraved germs, then h + g : D^{n} × D^{k} ⟶ R is neat with respect to a metric on D^{n} × D^{k} which restricts on a neighbourhood of (0, 0) ∈ D^{n} × D^{k} to a metric α_{1} + α_{2}, with α_{i} riemannian metrics on D^{n} and respectively D^{k} for i = 1, 2. For such a metric we have A_{h+g} ≃ A_{h} ∗ A_{g}.

b. A singular germ f = h + g with h and g non-depraved verifies the conclusions of Lemma 2.2 and of Proposition 2.3 for any metric of the type above.

Proof. Recall that a nondepraved germ f has the property:

∗∗ for sufficiently small ϵ, and for all x satisfying ||x|| ≤ ϵ, ∇f(x) = λx,

λ ∈ R, we have sign(λ) = sign(f) Our germ f = h + g continues to satisfy this condition for any metric α on D^{n} × D^{k} of the sort considered (this discussion is necessary because h + g might not be non-depraved). Indeed, let α_{i}, i = 1, 2 be as above. Whithout lack of generality we may assume that α_{1} and α_{2} are the standard metrics. Clearly, ∇^{α}(g + h) = ∇(h) + ∇(g). As g and h are both nondepraved and this condition is independent of the metric we get that if (z, y) ∈ D^{n} × D^{k} sufficiently close to (0, 0) and (∇(h)(z), ∇(g)(y)) = λ(z, y) , then sign(λ) = sign(g(y)) = sign(h(z)) = sign(h + g)(z, y). Assume now that the claim at a. is false. It follows that for any n ∈ N there is a point (x_{n}, y_{n}) such that {(x_{n}, y_{n})} converges to (0, 0), ∇^{α}(h + g)(x_{n}, y_{n}) = λ_{n}(x_{n}, y_{n}), for some λ_{n} ∈ R and (h + g)(x_{n}, y_{n}) = 0. In particular, we have ∇h(x_{n}) = λ_{n}x_{n}, ∇g(y_{n}) = λ_{n}y_{n}. This means sign(h(x_{n})) = sign(g(y_{n})) for sufficiently big n and contradicts h(x_{n}) + g(y_{n}) = 0. The homotopy equivalence A_{h+g} ≃ A_{h} ∗ A_{g} follows easily as in [2] or [11].

b. By inspecting the proof of Lemma 2.2 we see that, besides the neatness of the singular germ, the only property that is used is ∗∗. Thus, the statement of 2.2 remains valid for h + g with respect to any metric on D^{n} × D^{k} that is of the type discussed. The proof of Proposition 2.3 does only use this fact (besides neatness) and therefore also carries over to h + g.

Remark 4. a. Whenever talking about the sublink of a sum h + g as above we will assume the metric used to define this sublink to be as in 2.4 a. Notice that for a non-depraved germ the sublink is independent of the metric.

b. The homotopy equivalence at point a. has been also discussed in [3]. We point out that all the constructions of that paper work also for neat singularities. However, there is an imprecision in the statements 3.1 and 3.3 there. One should in fact assume that the functions involved have locally analytic or at least non-depraved singularities.

3. The sublink and PS deformations.

We need some elements of Conley index theory that we now recall [1][19]. Consider a continuous flow γ : X × R ⟶ X , X being a locally compact, metric space. Let S ⊂ X be a compact invariant set which is isolated in the sense that there is a compact neighbourhood N of S such that S ⊂ Int(N) and S is the maximal invariant set of γ inside N. Such a neigbourhood is called an isolating neighbourhood of S. A pair (N_{1}, N_{0}) of compact sets in N is an index pair for S in N if N_{0} ⊂ N_{1}, N_{1} − N_{0} is a neighbourhood of S, S is the maximal invariant set in the closure of N_{1} − N_{0}, N_{0} is positively invariant in N_{1} and, if for x ∈ N_{1} there is some T ≥ 0 such that γ_{T} (x) ∉ N_{1}, then there exists a τ > 0, τ < T with γ_{t}(x) ∈ N_{1} for 0 ≤ t ≤ τ and γ_{τ} (x) ∈ N_{0}. There are index pairs inside any isolating neighborhood of S. The Conley index of S, c_{γ}(S), is the homotopy type of the quotient space N_{1}/N_{0}. It is independent of the choices of the index pair. Moreover, suppose γ^{λ} : X × R ⟶ X is a family of flows depending continuously on the parameter λ ∈ [0, 1]. Assume S is an isolated invariant set of γ viewed as a flow on X × [0, 1]. Then S_{λ} = S ⋂ X × {λ} is an isolated invariant set of γ^{λ} and c_{γ}(S) = c_{γ^{λ}}(S_{λ}) for all λ ∈ [0, 1]. This last property of the Conley index is refered to as invariance to continuation (the invariant set S_{0} of γ^{0} being continued to the invariant set S_{1} of γ^{1}).

We will make use of the following definition.

Definition 2. Fix a riemannian metric β on R^{n} × [0, 1]. We denote by β_{t} the metrics on R^{n} given by β|_{R^{n}×{t}}, t ∈ [0, 1]. Let F : R^{n} × [0, 1] → R and let F_{t} = F |_{R^{n}×{t}}. A β-Palais-Smale deformation is a smooth function F as above such that ∇^{β_{t}}F_{t} induces a flow, the union of the critical sets of F_{t} for all t ∈ [0, 1] is compact and for any m ∈ R, a sequence (x_{n}, τ_{n}) ∈ R^{n} × [0, 1] that verifies ∇^{β_{τn}} F_{τ_{n}}(x_{n}, τ_{n}) → 0, |F (x_{n}, τ_{n})| ≤ m contains a convergent subsequence.

Two smooth, singular germs f, g : D^{n} ⟶ R are related by a Palais-Smale deformation if there is a metric β and a β- PS deformation F such that F_{0} has a single critical point and extends f, F_{1} has a single critical point and extends g.

We say that the germs f and g are related by a stable PS deformation if there is some non-degenerate quadratic form q : R^{k} ⟶ R such that f + q, g + q : D^{n} × D^{k} ⟶ R are related by a β-PS deformation with β_{t} a product metric in a neighbourhood of (0, 0) ∈ D^{n} × D^{k} for all t ∈ [0, 1].

Remark 5. The simplest example of a β - Palais-Smale deformation is that of a function F : R^{n} × [0, 1] ⟶ R that has the property that there is some ϵ > 0 such that for some fixed compact set K ⊂ R^{n} we have ||∇^{β_{t}}F_{t}(x)|| ≥ ϵ for all t ∈ [0, 1] and x ∉ K and ∇^{β_{t}}F_{t} induces a flow. All the PS deformations that will be constructed below are of this type.

3.1. Reduction to homotopy theory. Denote by ≃ homotopy equivalence and by ≃_{S} stable homotopy equivalence (two finite CW -complexes X, Y verify X ≃_{S} Y iff for some q ∈ N we have Σ^{q}X ≃ Σ^{q}Y).

Theorem 3.1. a. If two neat germs f and g are related by a PS deformation, then ΣA_{f} ≃ ΣA_{g}.

b. Assume that for two non-depraved germs f and g we have A_{f} ≃_{S} A_{g}. Then f and g are related by a stable PS deformation.

Proof. Assume that f is a neat germ and that γ, U(ϵ, δ) and A(ϵ, δ) are as in Lemma 2.2. The pair (U(ϵ, δ), A(ϵ, δ)) is an obvious index pair of 0 viewed as an isolated invariant set of γ. Therefore, as U(ϵ, δ) is contractible and A(ϵ, δ) ≃ A_{f} (by Remark 2), the corresponding Conley index verifies c_{γ}(0) ≃ ΣA_{f}.

Let β be a riemannian metric on R^{n}×[0, 1] and let F : R^{n}×[0, 1] ⟶ R be a β-PS deformation relating f to g. Now, f might not be neat with respect to β_{0}. However, one can relate β_{0} to the canonical metric by a path of metrics β′ with β′_{0} = β_{0} and β′_{1} the canonical metric. A look to the induced corresponding induced gradient flows shows by continuation that the Conley index of 0 with respect to the β_{0}-gradient flow of f coincides with ΣA_{f}. A similar argument shows that the Conley index of 0 with respect to the β_{1} gradient flow of g coincides with ΣA_{g}.

Denote by γ^{τ} the flow induced by the negative of the β_{τ} gradient of F_{τ}, −∇^{τ} F_{τ}. The restriction imposed on the family F_{τ} implies that there is a compact set Q such that for all τ the critical set of F_{τ} is included in Q. Let T = max{|F (x, t)| : x ∈ Q, t ∈ [0, 1]}. Moreover, there exists a compact set K ⊂ R^{n} and a constant ϵ > 0 such that the gradient of F_{τ} has β_{τ} norm bigger than ϵ for x ∉ K, |F_{τ} (x)| ≤ 2T, ∀τ. Consider two points a, b ∈ Q. For a fixed τ assume there are t_{0} and t_{1} and x such that γ_{t_{0}}^{τ}(x) = a and γ_{t_{1}}^{τ}(x) = b. We have 2T ≥ |F (a) − F (b)| = | ∫_{t_{0}}^{t_{1}} \frac{d}{dt}(F_{τ} (γ_{t}^{τ} (x)))dt| = | ∫_{t_{0}}^{t_{1}} β_{τ} (∇^{τ} F_{τ} , \frac{dγ^{τ}}{dt})dt| ≥ ϵ^{2}H_{a,b}^{τ} where H_{a,b}^{τ} is the time that the flow line passing through a is spending out of K and before reaching b. The compacity of Q and this uniform bound for the times H_{a,b}^{τ} implies that there is another compact K′ such that for any τ, a flow line of γ^{τ} starting in Q and returning in Q remains in the meanwhile in K′. (If this would not happen, then there would exist sequences x_{n} ∈ Q, t_{n} ∈ R such that lim(x_{n}) = x_{0}, lim(t_{n}) = t_{0} and ||γ_{t_{n}}^{τ}(x_{n})|| → ∞. But this would imply ||γ_{t_{0}}^{τ}(x_{0})|| = ∞.) As the critical points of F_{τ} are inside Q this means that K′ × [0, 1] is an isolating neighbourhood for the maximal invariant set of the flow γ. By the continuation property of the Conley index we have that the Conley index of the maximal invariant set of γ^{0} coincides with the Conley index of the maximal invariant set of γ^{1}. However, it was noticed above that the first Conley index is, up to homotopy, ΣA_{f} and the second is ΣA_{g}.

Remark 6. The use of the Conley index in similar settings is relatively standard. Another such application appears in [6]. The relation between the Conley index of an isolated critical point and its sublink appears in [2]. I do not know how to prove the invariance of the homotopy type of the suspension of the sublink whithout using the Conley index.

The second part of the theorem follows from the next result.

Proposition 3.2. If two non-depraved germs f, g : D^{n} ⟶ R have the property that A_{f+q} and A_{g+q} are isotopic in S^{n−1} for some non-degenerate quadratic form q : D^{k} ⟶ R, then they are related by a stable PS deformation.

Assuming this we proceed as follows. By hypothesis there is a k ∈ N such that Σ^{k}A_{f} ≃ Σ^{k}A_{g}. Let q : R^{2k+n+2} ⟶ R be a nondegenerate quadratic form of index k. Of course, q is non-depraved. Therefore, A_{f+q} ≃ A_{f} ∗ A_{q} ≃ Σ^{q}A_{f} ≃ Σ^{q}A_{g} ≃ A_{g+q} by Lemma 2.4. This means that A_{f+q} and A_{g+q} are two thickenings in S^{2k+2n+1} of a CW complex of dimension at most n+k−1. By results of [23][10] these two thickenings are isotopic in S^{2k+2n+1} and by applying 3.2 we obtain that f and g are related by a stable PS deformation.

To end the proof we need to show 3.2. A first step is given by the next statement.

Lemma 3.3. a. If A ⊂ S^{n−1} is a compact n − 1-manifold with boundary, then there is a non-depraved germ f with A_{f} = A.

b. Consider three non-depraved germs f : D^{n} ⟶ R, h : D^{k} ⟶ R and g : D^{n+k} ⟶ R. If A_{f+h} = A_{g}, then f + h and g are related by a PS deformation.

Proof of 3.3. The first part of the lemma is similar to a result in [2] and is inspired by a general technique for constructing isolated critical points that appears in [21]. Here is the construction. Let ϕ : [0, 1] ⟶ [0, 1] be a C^{∞} function that satisfies: dϕ/dt(t) > 0 if t ≠ 0; d^{n}ϕ/dt^{n}(0) = 0 for all n ∈ N; ϕ(1) = 1. Let g : S^{n−1} ⟶ R be a C^{∞} function having 0 as a regular value and such that g^{−1}((−∞, 0]) = A. Define f : D(δ, 0) ⟶ R by f(y) = ϕ(||y||^{2})g(x) where y = (x, ||y||) ∈ S^{n−1} × [0, 1] in polar coordinates. As all the derivatives of ϕ vanish at 0 it follows that f is C^{∞}. Clearly, f^{−1}((−∞, 0])) ⋂ S(δ, 0) = A. Now, in polar coordinates (x, t), the gradient of f has the form ((ϕ)(t^{2})/t∇g, 2t(dϕ/dt)(t^{2})g). This means that the only singularity of f is at 0. Moreover, whenever g ≠ 0 the product of this gradient with the position vector has sign = sign(g) = sign(f). Together with the fact that if x ∈ f^{−1}(0), x ≠ 0, then the vector x belongs to T_{x}(f^{−1}(0)) this implies that 0 is a non-depraved critical point of f.

For the second part we first assume k = 0 = h. Notice that if f : D^{n} ⟶ R has a single singularity at 0 which is non-depraved, then there is a new smooth function \overline{f} with a single critical point and small disks (all centered at 0) D′ ⊂ D′′ ⊂ D^{n} such that f = \overline{f} on D′ , sign(f) = sign(\overline{f}) on D′′ and sign(∇\overline{f}(x) • x) = sign(\overline{f}) for x ∈ ∂D′′. Indeed, consider a cylindrical neighbourhood U(ϵ, δ) as in Lemma 2.2. We may round the corners of U(ϵ, δ) getting a new neighbourhood U′ in such a way that −∇f points in on ∂U′ ⋂ f^{−1}((0, ∞)) and out on ∂U′ ⋂ f^{−1}((−∞, 0)). We have, as in 2.2, a diffeomorphism of pairs (U′, A(ϵ, δ)) ≈ (D′′, A_{f}) for some small disk D′′ containing U′. Moreover, this diffeomorphism can be taken to be equal to the identity on some small disk D′ ⊂ U′. We define \overline{f} as the composition of the restriction of f to U′ with the inverse of this diffeomorphism.

We perform the same construction for g thus getting a function \overline{g}. We have that −∇\overline{g} and −∇\overline{f} point inside (respectively outside) D′′ precisely on A_{−f} = A_{−g} (respectively A_{f} = A_{g}). This implies that the function G_{τ} = τ\overline{f} + (1 − τ)\overline{g} will have the same properties on ∂D′′. In particular its gradient will never vanish on ∂D′′ for any τ ∈ [0, 1]. Because of this we may use the same blow up technique as in the proof of the first part of 2.3 to extend the function G : D′′ × [0, 1] ⟶ R to a smooth function F : R^{n} × [0, 1] ⟶ R such that F_{τ} extends G_{τ}, with respect to a suitable metric F is a Palais-Smale deformation such that F_{0} extends f and F_{1} extends g and the critical points of F_{τ} coincide with those of G_{τ}. Notice that we may assume that the relevant metrics restrict to the canonical one in a neighbourhood of 0 ∈ R^{n}.

We come now back to the general case when k > 0 and h ≠ 0. Because of Lemma 2.4, sums of non-depraved singularities are neat with respect to metrics restricting to products in a neighbourhood of the origin and have cylindrical neighbourhoods satisfying the diffeomorphism of Lemma 2.2. Therefore, the construction above can also be applied to f + h. The statement follows.

Proof of 3.2. Let h : A_{f+q} ×[0, 1] ⟶ S^{n−1} ×[0, 1] be the smooth embedding provided by the isotopy of A_{f+q} and A_{g+q}. It has the properties that h_{t} is an embedding for all t ∈ [0, 1], h_{0} is the inclusion of A_{f+q}, and h_{1} is a diffeomorphism A_{f+q} ≈ A_{g+q}. Denote by A the image of h. We may use a parametrized version of the construction in Lemma 3.3. a. to obtain a function G : D^{n} × [0, 1] ⟶ R such that for each t ∈ [0, 1] the function G_{t} has a single singularity at 0 which is non-depraved. Of course, the sublink of G_{0} is A_{f+q} and that of G_{1} is A_{g+q}. A parametrized version of Proposition 2.3 shows that G can be extended to a PS deformation of G_{0} to G_{1}. By applying the second part of Lemma 3.3 we conclude the proposition.

Corollary 3.4. Under the assumptions of the point a. of the theorem we also have ΣL_{f} ≃ ΣL_{g}.

Proof. From the proposition we obtain that L_{f} is the void set if and only if L_{g} is void. We now assume that L_{f} is not void. We have an obvious homotopy push out:

\xymatrix{L_{f} \ar[d] \ar[r] & A_{f} \ar[d]_{i} \\ A_{−f} \ar[r]_{j} & S^{n−1}}

The maps i and j are null-homotopic. This implies that up to homotopy ΣL_{f} ≃ ΣA_{f} ∨ ΣA_{−f} ∨ S^{n−1}. By applying the theorem we have ΣA_{f} ≃ ΣA_{g} and also ΣA_{−f} ≃ ΣA_{−g}.

Corollary 3.5. The relation ”f is related to g by a stable PS deformation” is an equivalence relation on the class of non-depraved germs.

Proof. Obviously, our relation is well defined, reflexive and symmetric. The only difficulty is transitivity. Assume that f is related to g by a stable PS deformation and that g is related to h by another such deformation. Then A_{f} ≃_{S} A_{g} ≃_{S} A_{h} by the first point of the theorem. But now point b. implies that h and f can be related by a stable PS deformation.

Remark 7. Recall that the standard equivalence relation for singularities (for example see [12]) is right equivalence. Two singularity germs f and g are right equivalent if there is a diffeomorphism germ h : R^{n} ⟶ R^{n} and such that f = g ∘ h. Obviously, if f and g are non-depraved and right equivalent, then A_{f} ≈ A_{g} and therefore f and g are stably PS equivalent. It is alo worth mentioning that right equivalence is not implied only by the existence of an isotopy of the inclusions A_{f} ⊂ S^{n−1} and A_{g} ⊂ S^{n−1} [12].

4. The main theorems and Hamiltonian flows.

4.1. Proof of Theorem 1.1. Let Sing_{n}^{P S} be the set of stable PS equivalence classes of non-depraved germs f : D^{n} ⟶ R. Let Ψ : Sing_{n}^{P S} ⟶ CW_{n} be defined by Φ([f]) = [A_{f}] where [f] is the stable PS equivalence class of f and [X] is the stable homotopy equivalence class of the complex X ⊂ S^{n−1} (CW_{n} is the set of stable homotopy classes of complexes admitting a thickening in S^{n−1}). By Theorem 3.1 and Lemma 3.3 we already know that Φ is a bijection. The map Φ is clearly compatible with the involution f → −f as A_{−f} is the closure of the complement of A_{f} in S^{n−1}. To end the proof of Theorem 1.1 we only need to discuss the two operations which are transported by Φ to the wedge and, respectively, the join of CW -complexes. The existence of these two operations is obvious as they can be defined using the bijection Φ^{−1}. However, it is useful to have a definition that is more geometric.

Fusion of singularities. Given two non-depraved germs f, g : D^{n} ⟶ R we construct a function k in the following way. Let U = U(ϵ, δ) and U′ = U′(ϵ, δ) be cylindrical neighbourhoods for f and g. We take the connected sum of U and U′ by first identifying a disk in D′ ⊂ f^{−1}(0) ⋂ ∂U with a disk D′′ ⊂ g^{−1}(0) ⋂ ∂U′ and then extending this identification to the set W of points in ∂U and points in ∂U′ situated on flow lines that cross D′ respectively D′′. Of course, W is again a disk (of dimension n − 1) and the union U ⋃_{W} U′ is a disk of dimension n. We define the function k : U ⋃ U′ ⟶ R by pasting together f and g on W. Of course by composing with a diffeomorphism we may assume that k is defined on D^{n}. We see that k has exactly two critical points both with the same critical value equal to 0, that it is negative or zero on a set ambiently isotopic to the connected sum of A_{f} and A_{g} and that it extends (translations) of both f and g.

Assume now [f] and [g] are in Sing_{n}^{P S}. Let [f] ∨ [g] ∈ Sing_{n}^{P S} be the stable PS equivalence class of a non-depraved germ h : D^{n} ⟶ R which admits a PS deformation H such that H_{0} extends h and H_{1} extends the function k constructed above and H_{0} as well as H_{1} do not have more critical points than h, respectively k. By a method similar to the proof of 3.1 a. we see that ΣA_{h} ≃ ΣA_{f} ∨ΣA_{g} (this is because the Conley index of the maximal invariant set of k is precisely this wedge). As Φ is bijective it follows that this operation is well defined once we prove that such functions h, H exist. This follows immediately by the same methods as in Proposition 3.2 (the only thing that is crucial is that even if k has two critical points, it still admits cylindrical neighbourhoods homeomorphic to disks; in this case they correspond to connected sums of cylindrical neighbourhoods of f and g).

Exterior sum. For [f] ∈ Sing_{n}^{P S} and [g] ∈ Sing_{k}^{P S} let [f] ⊕ [g] = [h] ∈ Sing_{n+k}^{P S} where h is a non-depraved germ such that there is a PS deformation relating h to f +g (such a germ h exists by Lemma 3.3). By 2.4 for a metric restricting to a product in a neighbourdood of the origin we have that f +g is neat and A_{f+g} ≃ A_{f} ∗ A_{g} = A_{h}. Assume h′ is any other non-depraved germ with the same property relative to f + g. Then, A_{h′} ≃_{S} A_{h} and therefore h and h′ are stably PS equivalent. As a consequence the operation is well defined.

As an immediate consequence of the properties of these two operations we obtain Theorem 1.1.

4.2. Proof of Theorem 1.2 and of Corollary 1.3. For completeness, we start with by recalling the basic notions needed.

A symplectic form ω on R^{2n} is a 2-form that is closed and nowhere degenerate. Assume α is some riemannian metric on R^{2n}. As before, for a smooth function g : R^{2n} ⟶ R let ∇^{α}g be the α - gradient of g. The Hamiltonian vector field, H_{g}, induced by g, is defined by the equation ω(H_{g}, X) = dg(X) that holds for all smooth vector fields X. It has the property that ∇^{α}g and H_{g} are α-orthogonal. Therefore, H_{g} is tangent to the hypresurfaces g^{−1}(a).

The Hamiltonian flow induced by g and ω is the flow obtained by integrating H_{g}.

We recall that we assume our isolated singularities to be different from local extrema.

If f is not self-dual, then it has bounded orbits. The idea of the proof is simple and similar to §I. 9.2 of [1]: assuming that f does not have bounded orbits we will show that there exists a continuation from the gradient flow of \overline{f} to the gradient flow of −\overline{f} in such a way that there is some global, compact, isolating neighbourhood (here \overline{f} is an arbitrary PS-extension of f). This implies, by the continuation properties of the Conley index that ΣA_{f} ≃ ΣA_{−f} which shows by Theorem 1.1 that f and −f are stably PS equivalent. It is useful to note that the intermediate stages of the continuation that will be constructed are not gradient flows and, therefore, we do not construct directly a PS deformation of f to −f.

Fix a metric α on R^{2n}. Let \overline{f} be some PS extension of f (relative to α). Fix also a symplectic form ω on R^{2n}.

Let ϕ, ψ, ν : [−1, 1] ⟶ [0, 1] be smooth functions such that ϕ(x) = 0 for x ≥ 0, ψ(x) = 0 for x ≤ 0, ν(x) = 0 for |x| ≥ 1/2, ϕ is decreasing, ϕ(−1) = 1, ψ is increasing, ψ(1) = 1; ν is increasing for x < 0, decreasing for x > 0, ν(0) = 1; ϕ(x) + ψ(x) + ν(x) = 1 for all x ∈ [−1, 1]. Let X = −∇^{α}\overline{f} and let H = H\overline{f} be the Hamiltonian vector field induced by \overline{f} and let h be the associated Hamiltonian flow.

Define a new vector field on [−1, 1] × R^{2n} by V (t, x) = ϕ(t)X(x) + ν(t)H(x) − ψ(t)X(x). Let γ be the associated flow on [−1, 1] × R^{2n} and let γ^{τ} be the restriction of γ to {τ} × R^{2n}.

For a flow η and a subset K of its domain we denote by I_{η}(K) the maximal invariant set of η inside K.

We consider the set N = [−1, 1] × D(R, 0). This set is certainly compact. Notice that I_{γ}(N) = [−1, 1] × {0} ⋃ I_{h}(D(R, 0)). This happens because

I_{γ^{τ}} (D(R, 0)) = {0} if τ ≠ 0 (1)

and γ^{0} = h.

Assume that for all x ∈ S(R, 0) the h-orbit of x is not bounded. This is equivalent to the fact that I_{h}(D(R, 0)) ⋂ S(R, 0) = ∅. It follows that I_{γ}(N) ⊂ Int([−1, 1]×D(R, 0)) and therefore [−1, 1]×D(R, 0) is an isolating neighbourhood of this invariant set. The continuation properties of the Conley index, the identity (1) and the particular form of the Conley index of a neat singularity imply that ΣA_{f} ≃ c_{γ^{−1}}(0) = c_{γ^{1}}(0) ≃ ΣA_{−f}.

Remark 8. Clearly, the same argument as above shows that, if U(R), R ∈ R^{+}, is any family of compact neighbourhoods of 0 with mutually disjoint boundaries and such that R < R′ implies U(R) ⊂ U(R′) and ⋃ U(R) = R^{n}, then there is a bounded orbit in U(R) that intersects ∂U(R).

Group structure on Sing^{T}. We will prove here Corollary 1.3.

We first verify that tame equivalence is indeed an equivalence relation. Recall that we say that two non-depraved germs f and g are tamely equivalent if there are self-dual, non-depraved germs T_{1} and T_{2} and quadratic forms q_{1} , q_{2} such that f + T_{1} + q_{1} and g + T_{2} + q_{2} are related by a PS deformation. By Theorem 3.1 this is equivalent to A_{f} ∗ A_{T_{1}} ≃_{S} A_{g} ∗ A_{T_{2}}. Notice that if T_{i} is a non-depraved germ which is self-dual for i = 1, 2, then T_{1} ⊕ T_{2} is self-dual. This immediately implies that our relation is an equivalence.

By Theorems 1.2 and 1.1, the statement is now obvious. Indeed, we have that Sing^{T} is isomorphic, via Φ, to CW_{∗}^{S} factorized by the equivalence relation identifying X to ΣX for every CW -complex X and, additionally, killing all homotopy types of sublinks of self-dual germs. All these germs correspond via Φ to self -dual complexes. Moreover, as germs of the form f ⊕ (−f) are self-dual (as they have a sub-link of the form A ∗ A^{∗} with A^{∗} the Spanier-Whitehead dual of A) we obtain that Sing^{T} is indeed a group and the required isomorphism is immediate.

If g is a non-depraved germ in the stable PS class of [f] + [q] , then A_{g} ≃_{S} A_{f} and as \overline{Φ}(f) ≠ 0 it follows that A_{g} is not (Spanier-Whitehead) self-dual, hence g has bounded orbits.

4.3. Closed Orbits. In this subsection we will show Corollary 1.4 and discuss its relations with results in the literature.

As above, we assume a symplectic form ω fixed on R^{2n}. Recall that by closed orbit of a flow γ we mean either an orbit generated by a point x such that there is a period T ∈ R with γ_{T} (x) = x or the closure of an orbit generated by a point x such that the two limits lim_{t→+∞} γ_{t}(x) and lim_{t→−∞} γ_{t}(x) exist and are equal. Recall also the ω-limits of a point x in the flow γ: ω^{+}(x) = ⋂_{t>0} \overline{γ_{[t,∞)}(x)}, ω^{−}(x) = ⋂_{t>0} \overline{γ_{(−∞,−t]}(x)}.

We start with a particular case.

Lemma 4.1. Generically, any PS function g : R^{2n} ⟶ R with a single critical point which is non-degenerate of index different from n induces a Hamiltonian flow with at least one non-constant closed orbit.

Proof. From Theorem 1.2 we know that the function g has bounded orbits. To show that any generic such g induces a Hamiltonian flow with at least one non-constant closed orbit, we are going to use the C^{1} closing lemma of Pugh and Robinson [15] in the following particular form ( see 11.3 of [15]): there is a dense set of Hamiltonian vector fields, S, on R^{2n} that satisfy Ω_{c}(S) ⊂ \overline{Γ}(S). Density is understood here in the C^{1} Whitney strong topology; \overline{Γ}(S) is the closure of the set of points on periodic trajectories; Ω_{c}(S) is the set of non-wandering points with at least one non-void ω-limit. We make here a distinction between closed orbits and periodic trajectories: the first class contains the second but are added the orbits that close at a stationary point. Assume that the index of the critical point of g is k. There is a C^{2} (Whitneycompact open topology) neighbourhood of g consisting of functions that have a single critical point which is non-degenerate and of index k. Therefore, generically, we may assume that g induces a Hamiltonian vector field H_{g} that satisfies Ω_{c}(H_{g}) ⊂ \overline{Γ}(H_{g}). We may also assume that the critical point of g is 0. Assume that x ∈ R^{2n} generates a bounded orbit of the flow induced by H_{g}. There are two possibilities, either ω^{+}(x) = ω^{−}(x) = 0 and in this case the orbit of x closes at 0 or there is a point y ∈ ω^{+}(x) ⋃ ω^{−}(x) different from 0. This point is non-wandering and notice that the orbit generated by y is also bounded and hence its ω-limits are non empty. Therefore, y is in the closure of the space of closed orbits and therefore this space contains more than the point 0.

Proof of Corollary 1.4. Let f : D^{2n} ⟶ R be a non-depraved germ and let \overline{f} be any PS extension of f. Close to \overline{f} we may find a generic family of smooth functions g such that the induced Hamiltonian vector fields are C^{1} generic in the sense above when restricted to a closed disk D and g is a Morse-Smale function with critical points ∈ D of distinct critical values (this happens because these Morse functions form an open and dense set). Assume a_{1}, ..., a_{m} are the critical values corresponding respectively to the critical points x_{1}, ..., x_{m}. We may apply Lemma 4.1 to each critical point x_{i} at a time. Indeed, this lemma depends on the existence of bounded orbits of the Hamiltonian flow inside f^{−1}([a_{i} − ϵ, a_{i} + ϵ]) for ϵ small and all i. But the same argument as that used in the first step of the proof of Theorem 1.2 can be applied in this situation (see Remark 8) and it leads to the existence of closed orbits inside this set whenever the index of the critical point x_{i} is not n (or in other words whenever x_{i}, as singularity, is not self-dual). It is easy to see that the number of critical points of index k of g is bounded from below by rk(H_{k−1}(A_{f}; Z)). Indeed, the maximal invariant set of the flow induced by the gradient of g, is compact and as g is very close to f by continuation its Conley index is ΣA_{f}. On the other hand the Morse complex of g computes the homology of this Conley index and our statement is implied by the Morse inequalities.

Remark 9. a. It is useful to recall that the problem of finding closed orbits on some hypersurface h^{−1}(a) for the Hamiltonian flow induced by a function h : R^{2n} ⟶ R depends only on the set h^{−1}(a) and not on h. This provides the connection between our result and, for example, that of Rabinowitz [16] claiming that if h^{−1}(a) is diffeomorphic to S^{n−1} under radial projection then it contains a periodic orbit. Indeed, if h^{−1}(a) has this property, then by possibly replacing h with a different function whithout modifying the preimage of a, we may assume that h has a single critical point which is non-depraved and a minimum. The Corollary 1.4 implies that generically we can find closed orbits on a hypersurface h^{−1}(a). This is much weaker than Rabinowitz’ result, of course. However, our result becomes of interest in the cases when the hypersurfaces in question are not compact and the singularity of h is complicated. In this non-compact setting, it appears that no other tools are available.

Of course, one would like to strengthen 1.4 by producing closed orbits whenever self-duality is not present whithout the genericity assumption. However, some additional conditions are probably necessary as there are compact hypersurfaces that do not carry any closed orbits [7].

b. Estimating precisely the number of critical points in close Morse approximations is rather subtle. The question turns out to depend on whether the approximation needs to be only C^{0} close or higher [11]. For a more general discussion on morsification see also [17].

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