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Improbability of Wandering Orbits Passing Through a Sequence of Poincar\'e Surfaces of Decreasing Size

Given a volume preserving dynamical system with non-compact phase space, one is sometimes interested in special subsets of its wandering set. One example from celestial mechanics is the set of initial values leading to collision. Another one is the set of initial values of semi-orbits, whose asymptotic velocity does not exist as a limit. We introduce techniques that can be helpful in showing that these sets are of measure zero: by defining a sequence of hypersurfaces, that are eventually hit by each of those semi-orbits and whose total surface area decreases to zero.


Introduction and Main Result
Let P d be a smooth manifold with a volume form Ω and a C 1 vector field X : P → T P , so that the Lie derivative L X Ω vanishes. By standard results of ordinary differential equations, the flow Φ associated to the differential equatioṅ x = X(x) uniquely exists on a maximal neighborhood D ⊆ R × P of {0} × P in extended phase space, Φ ∈ C 1 (D, P ), and Φ preserves the volume form Ω. The flow's domain of definition is of the form (1.1) O(x) := Φ ((T − (x), T + (x)), x) we denote the orbit passing through x ∈ P , and by O + (x) := Φ ([0, T + (x)), x) the semi-orbit. We consider the wandering set of Φ Wand ≡ Wand Φ ⊆ P, consisting of those x ∈ P which have a neighborhood U x so that for a suitable time t − ∈ (0, T (x))

Remark 1.1 (Wandering set)
1. In view of applications (see Example 1.6 below) we allowed for finite escape times T ± (x). So Φ does not in general define an R-action on P . In this sense our notion of 'wandering set' is a generalization of the usual definition.

As T is lower semi-continuous, the set of singular points
is a Borel set. It is wandering for the same reason, see Lemma 1.2 below.
3. Trivially, equilibrium points are nonwandering, so that The latter is an open submanifold of P because of X ∈ C 1 (P, T P ). Thus, we assume the vector field X to be non-vanishing on P from the outset, without loss of generality. ✸ Lemma 1.2 Sing ⊆ Wand .

Assumptions:
1. The vector field X is transversal to their relative interior ι m : H m → P . Thus Then we denote the set of transition points, whose forward orbits eventually hit all of these surfaces by Our main result is the following.
Theorem A From the assumptions it follows that Ω(Trans ∩ Wand) = 0.

Remark 1.3 (Theorem A)
3. Finally, the symplectic manifold (P, ω) may be Kähler, that is, equipped with a Riemannian metric g and complex structure J, so that ω(X, Y ) := g(JX, Y . We assume that for some α ∈ (0, 2), and, say lim q →∞ V (q) = 0. Then for all x 0 ∈ Sing we have lim t→T ( for all sufficiently large m ∈ N. The symplectic manifold (T * (R n \{0}), ω 0 ) is Kähler. Using Theorem B, for F m := {q ∈ R n | q = 1/m} with the Riemannian volume element dF m . Here Q(q) is the projection along the direction q/ q . The expression is derived as follows: • The factor vol(S n−1 ) in (1.6) is the Riemannian volume of a unit sphere in momentum space.
• The factor 1/2 is due to the condition p, q ≤ 0 in the definition (1.5) of the surface H m .
• At the point q on the sphere F m of radius 1/m, the corresponding half-sphere im momentum space has radius 2(E − V (q)). For ∇V (q) parallel to q this would lead to a Riemannian volume element (2(E − V (q))) (n−1)/2 dF m (q).
• By Pythagoras, this is multiplied by the ratio 1/2 between hypotenuse and adjacent side for the direction of the gradient.
If we additionally assume that the force is radial, that is Q∇V (q) = 0, then it follows that the singular set has measure zero. On the other hand, it is well known that collision occurs with positive measure if V (q) = − q −α and α ≥ 2. So in the example the criterion is optimal.
If (P, ω) = (T * M, ω 0 ), with the canonical symplectic form ω 0 on the cotangent bundle of a Riemannian manifold (M, h) and if the Hamiltonian function is of the form 'kinetic+potential', then there is a useful integration formula, see Theorem C. This can be applied to our example.
Example 1.7 If we instead use Theorem A, employing Theorem C, we obtain, without assuming that Q∇V (q) = 0, The integral is bounded above by c 3 m (α−2)(n−1)/2 and thus goes to zero as m → ∞. So the singular set has measure zero. ✸ The outline of the article is as follows: Section 2 abstractly considers a discretized version of the dynamics. In Section 3, we show that indeed only the surface areas of the transverse parts H m of each H m are of interest, since hitting the boundaries ∂H m of the surfaces will be shown to be improbable. In Section 4, we make a crucial step in the proof of Theorem A by contradiction: if the set of wandering transition points had positive measure, then in one Poincaré surface we would find a compact subset, whose intersection with the set of these points had positive area. The main tool in doing so is a version of the Flow-Box Theorem, which we will state at the beginning of that section. Then, in Section 5 we will take a look at the progression of those transition points by defining Poincaré maps between the surfaces (hence the name Poincaré surfaces), which are shown to be area preserving. Since the total area of the surfaces is assumed to decrease to zero, this contradicts an initial set to have positive area.

Remark 1.8 (Transversality)
In Section 6, we show how the result can be restated if the symplectic manifold is Kähler. Finally, in Section 7 we indicate how we apply the scheme given here in forthcoming articles.

Discretization of the Problem
Here we consider a discrete dynamics. This will be used in Section 5 to model the dynamics on the Poincaré surfaces.
are assumed to have finite measures. The transition set is defined by Proof: Assuming the converse, we find k ∈ N and For such a k and U x we set K k := U x ∩ Trans ∩ Wand and As we have the disjoint union • But also µ(K k,ℓ ) = 0 for all ℓ ∈ N. For otherwise, using K k,ℓ ⊆ Trans, with

Then the map
is one to one, and thus µ( So we derived a contradicting to the assumption µ(Trans ∩ Wand) > 0. ✷ In Section 3 we will apply Lemma 2.1 to a Poincaré map for the flow Φ, on a certain subset of ∪ m∈N H m .

Transversality
In the following sections, we need to move points along their orbits, until they hit one of the Poincaré surfaces. We usually need the orbits to be transverse to the surfaces to ensure that the corresponding (local) Poincaré maps are C 1 . Also, we want to define a global continuous discrete dynamics for using Lemma 2.1. As we would like to exclude orbits hitting a boundary ∂H m from our considerations early on, we claim: The measure Ω(T ) vanishes for the set is of measure zero. ✷

Proof of Theorem A:
We now use Lemma 2.1 in the following way. As Set ∩T = ∅, the flow Φ is transversal to M.
• T : M → M is defined as the next intersection with a Poincaré surface. As Set ⊆ Trans, this exists. So the return time is finite. Since Φ is transversal to M, it is positive, and one sets Notice that by transversality T ∈ C(M, M). By definition T is injective.

Localization to one Poincaré Surface
As pointed out in the introduction, one step of proving Theorem A is the following implication: if Trans Φ had positive measure, then there would be some compact subset of some Poincaré surface, whose intersection with Trans Φ had positive area: Our main tool in proving this is the following Flow-Box Theorem, which identifies the change of volume of sets like Φ((−t, t)×K) (with an appropriate codimension one ∂-submanifold K) in ongoing time with the surface area of the submanifold K.
To be more precise:

Lemma 4.2 (Flow-Box Theorem)
Let H ⊆ P be a submanifold of codimension one, such that the vector field X is transverse to H. Then x ∈ H has a compact ∂-submanifold K ⊆ H as a neighborhood, L t := Φ((−t, t) × K) is a ∂-submanifold of P for sufficiently small t > 0, and for all continuous functions f on K; here, F is the continuation of f to L t that is constant along the flow lines, i.e.
3. The local representative of the vector field X at any point of the chart is e 1 = (1, 0, . . . , 0) ∈ R n .
So L t is a ∂-submanifold of P , and its closure L t is a submanifold with corners. By Property 3. we can write the volume form Ω on L ε as using the time coordinate t of the chart ϕ. Therefore As the Lie derivative L X F Ω = (L X F )Ω + F L X Ω vanishes by Def. (4.2), we have dF V = di X F Ω = 0, too. So (4.3) implies F Ω = d(ti X F Ω) = d(tF V), and Stokes' Theorem tells us that In the last equation we used that V = i X Ω vanishes on Φ((−t, t), ∂K). ✷ With this, we can give the proof of the preceding lemma: Proof of Lemma 4.1: To simplify notation, we use the flow invariant set is a compact subset of H m with x m ∈ K m , which just like H m is of codimension one in P . Then we find ε > 0, such that for every point in K m the flow exists at least on the time interval (−ε, ε); then is well-defined, see Figure 3. By a possible diminishment of ε, we have that for all continuous functions f on the compact set K m , where Ω is the canonical volume form on P ; for F , we adopt the notation from Lemma 4.2 (continuation of f , which is constant along the flow lines).
3 with respect to an arbitrary Riemannian metric g on P and the metric d induced by g Now let f n (n ∈ N) be a sequence of smooth functions on H m with compact where again F n is the continuation of f n , that is constant along the flow lines.

Transit between the Poincaré Surfaces
Below, we will identify points on one surface with certain elements on their positive semi-orbit, lying on another surface. We will call such a mapping Poincaré map, and show that they are volume preserving. Now we give the general notion of Poincaré maps as used within this paper. We first consider general (incomplete) flows and then specialize to the Hamiltonian flow Φ, and to its restriction to energy surfaces. Then there exist a neighborhood U 0 ⊆ H 0 of x 0 and a so-called hitting time τ ∈ C 1 (U 0 , R) with τ (x 0 ) = t 0 , such that on U 0 the Poincaré map ψ, is a diffeomorphism onto its image U 1 .
2. Now consider the case of the flow Φ on the manifold P generated by X and preserving the volume form Ω. Then the restrictions of V = i X Ω to H i are volume forms, and the Poincaré map, see Figure 4, is volume preserving.

Submanifolds of Kähler Manifolds
If there is an underlying metric on the manifold inducing the symplectic structure, then it may be more convenient from a technical point of view, to use the metric structure to estimate the symplectic volume of a codimension two submanifold instead of directly integrating the volume form i X H σ. In this section, we show that this is possible. Let (P, g, J) be a 2n-dimensional Kähler manifold, with Riemannian metric g and complex structure J, which induce a symplectic form ω by ω(X, Y ) := g(JX, Y ) (X, Y ∈ X (P )).
Assume 4 that E ∈ R is a regular value of H, such that Σ E is a codimension-1 submanifold (if non-empty) of P . Let (H E m ) m∈N be a sequence of submanifolds of Σ E of codimension one. Let dP be the Riemannian volume form on P induced by g; then dP equals the canonical volume form Ω as defined above. Further, let dΣ E and dH E m be the Riemannian volume forms on Σ E and H E m respectively, induced by the restrictions g Σ E resp. g H E m of g to Σ E resp. H E m . Then it is sufficient to calculate the Riemannian surface areas of the submanifolds, in order to apply the technique from above. The following two lemmas will be used in the proof of Theorem B.
Lemma 6.1 Let ı : N → P be a submanifold of codimension two, being the preimage of a regular value of a smooth map F ≡ (F 1 , F 2 ) : P → R 2 . Then dN being the Riemannian volume form on N induced by g.

Lemma 6.2 (Wirtinger's inequality)
On an R-vector space V with scalar product ·, · , complex structure J : V → V and Kählerian symplectic form ω one has the inequality Proof: We assume without loss of generality that X = 1. Then the formula follows from Bessel's inequality, as ω(X, Y ) = JX, Y , and the pair {X, JX} ⊆ V is an orthonormal system. ✷ Now we use Lemma 6.1 and 6.2 to prove Theorem B from Section 1, which we cite for convenience: By applying Lemma 6.1, we get Then |η| ≤ 1 follows from Lemma 6.2, applied to the tangent spaces T x P for x ∈ H E m . ✷ Finally we consider the metric g on the cotangent bundle π : T * M → M of a Riemannian manifold (M n , h), given at x ∈ T * M by g x ((Q 1 , P 1 ), (Q 2 , P 2 )) := h π(x) (Q 1 , Q 2 )+h −1 π(x) (P 1 , P 2 ) is an almost complex structure. 5 So the symplectic manifold (T * M, ω 0 ) with J and g(u, v) = ω 0 (u, Jv) is an almost Kähler manifold. For a Hamiltonian H ∈ C 2 (T * M, R) of the form a regular value E ∈ V (R) of V and a hypersurface F ⊆ M of finite Riemannian volume, we obtain a hypersurface in Σ E = H −1 (E) given by So H E is of codimension two in T * M. By localization, if necessary we can assume that F is orientable and denote by N : F → T F M a (continuous) unit normal vector field. Then (applying p ∈ T * q M to N ∈ T q M) both project diffeomorphically to their common image in T * F ⊆ T * M, via The cotangent bundle T * F carries the canonical symplectic form ω F . 5 i.e., a smooth (1,1) tensor field with J 2 = −1.

Theorem C With respect to the embeddings ı
with the Lebesgue measure v k of the k-dimensional unit ball.
Proof: To prove (6.4), we represent F as a zero set of a smooth function f , defined in a neighborhood U ⊆ M of a point q 0 of F . We additionally require that df (N(q)) = +1 (q ∈ F := F ∩ U). Next we multiply the phase space function π * f : T * U → R with the functions (depending on the parameter E) The resulting functions g ± : T * U → R define maximal Hamiltonian flows Γ ± : D → T * U. As g ± vanishes on T * F M, there the flow lines of Γ ± coincide with those of the Hamilton flow of f . This means that Γ ± acts on the fibers of π : T * F → F : Note that Γ ± t does not change the argument of the square root. For initial conditions (q, p) ∈ H ± E ∩ T * U, this square root is the modulus of the (initial) momentum component p(N(q)) in the normal direction. So on T * U the flow Γ ± 1 maps H ± E into the cotangent bundle T * F . Formula (6.4) then follows like in Remark 1.4.3, using that ω F = ı * T * F ω 0 for the embedding ı T * F : T * F → T * M. In the left hand side of (6.5) we wrote for simplicity ω 0 instead of (ı ± E ) * ω 0 . When we apply (6.4), we obtain We write ω ∧n−1 F (n−1)! in the form dF ∧ dG, where on the fiber of π : T * F → F over q, dG q denotes the Riemannian volume element with respect to h −1 q . Restricted to the fiber over q ∈ F the image n ± (H ± E ) is a ball of radius 2(E − V (q)) with respect to · h −1 q . Integrating out the fiber yields (6.5). ✷

Applications and Open Questions
In this paper, we have defined the set of transition points depending on a given sequence of Poincaré surfaces. In applications, the perspective is usually vice versa: one is interested in showing the improbability of a given subset of the wandering set. If the technique presented here is ought to be applied, the task then is to find an appropriate sequence of Poincaré surfaces, such that their total area decreases to zero and that the wandering orbits under considerations can be shown to be transition points to that sequence. One important model, where these techniques can be implemented, is the Nbody problem of celestial mechanics. A subset of singular orbits (1.2) which are of special interest are collisions: they are exactly those singular orbits, where every particle has a limit position in configuration space, as time approaches the singularity. In the upcoming paper [FK18], we show the improbability of collisions in the N-body problem that may include some fixed centers as well. More generally, the result holds for a wide class of two-body interactions, including the gravitational case of celestial mechanics, but also e.g. Coulomb fields in electrostatic motion. Due to the technique devised in the present article, these estimates are optimal, concerning the power law of the two-body interactions.
For the same family of two-body interactions, also non-collision singularities can be treated in certain situations: in the upcoming paper [Fle18a] we show the improbability of non-collision singularities in systems with two free particles and an arbitrary amount of fixed centers. In the upcoming paper [Fle18b] we show the improbability of non-collision singularities in the four-body system. As pointed out, the technique presented here can only be used to show the improbability of certain subsets of wandering orbits. However, in principle the given result can be applied to show the improbability of certain sets of nonsingular wandering orbits, e.g. those for which the asymptotic velocity does not exist as a limit.