On the Scattering Length Spectrum for Real Analytic Obstacles

Luchezar Stoyanov

1 Introduction

Let K be an obstacle in IR^{n} (n ≥ 3), i.e. a compact subset of IR^{n} with C^{∞} boundary ∂K such that Ω_{K} = \overline{IR^{n} \ K} is connected. A scattering ray γ in Ω_{K} is an unbounded in both directions generalized geodesic in the sense of Melrose and Sjöstrand [MS1], [MS2] (cf. also Sect. 24.3 in [H]). We denote by T_{γ} the sojourn (travelling) time of γ (cf. Sect. 2). If ω ∈ S^{n−1} is the incoming direction of γ and θ ∈ S^{n−1} its outgoing direction, γ will be called an (ω, θ)-ray. The scattering length spectrum (SLS) of K is by definition the family of sets of real numbers SL_{K} = {SL_{K}(ω, θ)}_{(ω,θ)} where (ω, θ) runs over S^{n−1} × S^{n−1} and SL_{K}(ω, θ) is the set of sojourn times T_{γ} of all (ω, θ)-rays γ in Ω_{K}. Thus, SL_{K} is a map which assigns to each pair of directions (ω, θ) a set SL_{K}(ω, θ) of non-negative real numbers. It is known (cf. [St2]) that for n ≥ 3, n odd, we have SL_{K}(ω, θ) = sing supp s_{K}(t, θ, ω) for almost all (ω, θ). Here s_{K} is the scattering kernel related to the scattering operator for the wave equation in IR×Ω_{K} with Dirichlet boundary condition on IR × ∂Ω_{K} (cf. [LP1], [M]).

It is a natural and important problem in inverse scattering by obstacles to get information about the obstacle K from its SLS. It follows from results of A. Majda [Ma] (see also Majda and Ralston [MaR]) and P. Lax and R. Phillips [LP2] that the convex hull K̂ of K can be recovered from SL_{K}. Consequently, in the class of convex obstacles (as well as in the class of connected obstacles with analytic boundaries), K is completely determined by its SLS. However, as an example of M. Livshits shows (cf. Ch. 5 in [M]), in general SL_{K} does not determine K uniquely.

Following [St3], we will say that two obstacles K and L have almost the same SLS if there exists a subset R of full Lebesgue measure in S^{n−1} × S^{n−1} such that SL_{K}(ω, θ) = SL_{L}(ω, θ) for all (ω, θ) ∈ R. Let Trap(Ω_{K}) be the set of trapped points of the generalized geodesic flow F_{t}^{(K)} in T^{∗}(Ω_{K}) (cf. Sect. 2) and let U be an open ball in IR^{n} containing both K and L. It was shown in [St3] (cf. Theorem 2.1 below) that if K and L have almost the same SLS and satisfy some nondegeneracy conditions, then the flows F_{t}^{(K)} and F_{t}^{(L)} are conjugate on their phase spaces minus the set of trapping points Trap(Ω_{K}) = Trap(Ω_{L}) by a time preserving conjugacy which is smooth and symplectic on an open dense subset. Using this conjugacy, some relationships between K and L (under various conditions) were derived in [St3].

In this paper we consider obstacles with real analytic boundary ∂K. This assumption implies that the normal curvature of ∂K does not vanish of infinite order and in particular (cf. [MS1]) the GHF F_{t}^{(K)} is well defined as a global flow. We will also assume that K satisfies the following non-degeneracy conditions: γ_{K}(σ) = {F_{t}^{(K)}(σ) : t ∈ IR} is a non-degenerate simply reflecting ray for almost all σ ∈ S^{∗}(IR^{n} \U) such that γ_{K}(σ) has at least one transversal reflection point, and for almost all (x, ξ) ∈ S^{∗}(∂K) the set {t : F_{t}^{(K)}(x, ξ) ∈ S^{∗}(∂K)} is connected. (The latter means that the ray γ_{K}(σ) either has one single point of tangency to ∂K or it has one single gliding segment on ∂K and no other tangencies.) We will denote by Kreg the class of ostacles K satisfying these non-degeneracy conditions. It is not difficult to see that Kreg is of second Baire category in the space of all obstacles with the C^{∞} Whitney topology^{1}.

We will denote by ∂K^{(ob)} the observable part of the boundary ∂K, namely ∂K^{(ob)} is the union of all connected components of ∂K that have a common point with at least one scattering ray in Ω_{K}. The obstacle K will be called observable, if ∂K = ∂K^{(ob)}. It should be stressed that these definitions are only apropriate for the real analytic case.

The main result in this paper is the following.

Theorem 1.1. Let K, L ∈ Kreg be obstacles in IR^{n} with real analytic boundaries and let K and L have almost the same SLS. If K is such that S^{∗}(C) \ Trap(Ω_{K}) is connected, then ∂K^{(ob)} = ∂L^{(ob)}. If in addition both K and L are observable, then K = L.

In general it is not clear how restrictive the condition about the connectivity of S^{∗}(C) \ Trap(Ω_{K}) is. Proposition 5.5 below shows that this condition is satisfied when K is a finite disjoint union of strictly convex domains with C^{∞} boundaries. Combining it with Theorem 1.1, one gets the following.

Corollary 1.2. Assume that K is a finite disjoint union of strictly convex domains with analytic boundaries and let L ∈ Kreg be such that K and L have almost the same SLS. Then K = L.

It should be remarked that even the case considered in the above corollary is non-trivial. When the number of convex components of K is large and they are ”densily packed” (in a sense), then the flow F_{t}^{(K)} has a lot of tangencies that make its study rather difficult even though the boundary ∂K is locally very simple. It is still unknown whether the statement of Corollary 1.2 remains true without assuming real analyticity of the boundary.

Most of the paper actually deals with obstacles with C^{∞} boundaries. In Sect. 3 we study the sets of points σ ∈ S^{∗}(IR^{n} \ U) that generate trajectories having tangencies to ∂K. It turns out that those σ that generate trajectories having tangencies of order higher than 1 form a relatively small subset G_{K}. Roughly speaking, G_{K} has codimension 2 in S^{∗}(IR^{n} \ U), so it does not divide topologically the space S^{∗}(IR^{n} \ U). Using this fact, in Sect. 4 we prove Theorem 1.1.

In Sect. 5 we begin with an observation which applies locally near any difractive tangent point (x_{0}, ξ_{0}) ∈ S^{∗}(∂K) and any K with C^{∞} boundary. Namely, we show that there exists a local strictly convex smooth hypersurface Y near x_{0} such that the unit normal field to Y consists of vectors that are tangent to ∂K on an open neighbourhood of x_{0} in ∂K. We use this fact in the case when K is a finite disjoint union of strictly convex domains with C^{∞} boundaries to show that the set T_{0} of ponts σ ∈ S^{∗}(Ω_{K}) generating trajectories with at least two different tangent points is relatively small. This implies (cf. Proposition 5.5) that S^{∗}(C) \ Trap(Ω_{K}) is connected. In fact, more precise information about the set Trap(Ω_{K}) is obtained. For example, if X is a local smooth convex surface in the interior of Ω_{K} with distinguished unit normal field ν_{X} and X̃ = {(x, ν_{X}(x)) : x ∈ X} is such that X̃ ∩ T_{0} has codimension at least 2 in X̃ (which is probably always the case but we do not prove it here), then dim(X̃ ∩ Trap(Ω_{K})) ≤ n − 3, and in fact dim(X̃ ∩ Trap(Ω_{K}) \ T_{0}) = 0. Here dim is the topological dimension. It should be mentioned that at least in the last relation dim cannot be replaced by the Hausdorff dimension dim_{H}; one can easily show that in many cases dim_{H}(X̃ ∩ Trap(Ω_{K}) \ T_{0}) > 0 and it is not easy to estimate dim_{H} from above for the sets that we deal with.

1It is very likely that the non-degeneracy of γ_{K}(σ) for almost all σ such that γ_{K}(σ) has at least one transversal reflection point follows automatically from the real analitycity of ∂K.

Acknowledgement. This paper was written during my visit to Max-Planck-Institut für Mathematik and the University of Bonn with the support of the Alexander von Humboldt Foundation. Thanks are due to these organizations and personally to Prof. Werner Müller for his support.

2 Preliminaries

Let K be an obstacle in IR^{n} with C^{∞} boundary ∂K. The following conditions guarantees (cf. [MS2]) that F_{t}^{(K)} is well-defined as a global flow: for each (x, ξ) ∈ T^{∗}(∂K) \ {0} if the normal curvature of ∂K at x vanishes of infinite order in direction ξ, then all points (y, η) sufficiently close to (x, ξ) are diffractive points (roughly speaking, this means that ∂K is convex at y in the direction of η). Denote by K the class of obstacles that have this property. Notice that if ∂K is real analytic, then K ∈ K.

We refer the reader to [MS1], [MS2] (or Sect. 24.3 in [H]) for the definition of the generalized Hamiltonian (bicharacteristic) flow on a symplectic manifold with boundary. In the case of scattering by an obstacle K, this flow is generated by the principal symbol of the wave operator in IR × Ω_{K}. It defines in a natural way a flow F_{t}^{(K)} : T_{b}^{∗}(Ω_{K}) \ {0} ⟶ T_{b}^{∗}(Ω_{K}) \ {0}, called the generalized geodesic flow. Here T_{b}^{∗}(Ω_{K}) = T^{∗}(Ω_{K})/ ∼ is the quotient space with respect to the following equivalence relation on T^{∗}(Ω_{K}): (x, ξ) ∼ (y, η) iff x = y and either ξ = η or ξ and η are symmetric with respect to the tangent plane to ∂K at x. For K ∈ K the flow F_{t}^{(K)} on T_{b}^{∗}(Ω_{K})\{0} is continuous ([MS2]). However, in this paper we prefer to work with F_{t}^{(K)} on T^{∗}(Ω_{K}) \ {0}. The image S_{b}^{∗}(Ω_{K}) of the unit cosphere bundle S^{∗}(Ω_{K}) under the natural projection is invariant with respect to F_{t}^{(K)}.

A point σ = (x, ω) ∈ S^{∗}(Ω_{K}) is called non-trapped if both curves {pr_{1}(F_{t}^{(K)}(σ)) : t ≤ 0} and {pr_{1}(F_{t}^{(K)}(σ)) : t ≥ 0} in Ω_{K} are unbounded. Otherwise σ is called a trapped point. Here we use the notation pr_{1}(y, η) = y and pr_{2}(y, η) = η. Denote by Trap(Ω_{K}) the set of all trapped points. As the example of M. Livshits (cf. [M]) shows, in general Trap(Ω_{K}) may have positive Lebesgue measure and non-empty interior in S^{∗}(Ω_{K}). Let σ = (x, ξ) ∈ S^{∗}(Ω_{K}) be such that γ_{K}(σ) = {pr_{1}(F_{t}^{(K)}(σ)) : t ∈ IR} is a simply reflecting ray, i.e. it has no tangencies to ∂K. We shall say that γ_{K}(σ) is non-degenerate if for every t ≠ 0 with pr_{1}(F_{t}^{(K)}(σ)) ∉ K the map IR^{n} ∋ y ↦ pr_{2}(F_{t}^{(K)}(y, ξ)) ∈ S^{n−1} is a submersion at y = x, that is its differential at y = x has rank n − 1.

Denote by K_{0} the class of all obstacles K ∈ K so that ∂K does not contain non-trivial open flat subsets and such that γ_{K}(σ) is a non-degenerate simply reflecting ray for almost all σ ∈ S^{∗}(IR^{n}\U) such that γ_{K}(σ) has at least one transversal reflection point. One can show that K_{0} is of second Baire category in K with respect to the C^{∞} Whitney topology in K. That is, for every K ∈ K, applying suitable arbitrarily small C^{∞} deformations to ∂K, one gets obstacles from the class K_{0} and in fact ”most” deformations have this property.

Set Ṫ^{∗}(Ω) = T^{∗}(Ω) \ {0} and denote by Ṫ_{b}^{∗}(Ω) the projection of Ṫ^{∗}(Ω) in T^{∗}(Ω)/ ∼.

The folowing theorem gives a necessary and sufficient condition for two obstacles in the class K_{0} to have almost the same SLS.

Theorem 2.1.([St3]) If the obstacles K, L ∈ K_{0} have almost the same SLS, then there exists a homeomorphism

Φ : Ṫ_{b}^{∗}(Ω_{K}) \ Trap(Ω_{K}) ⟶ Ṫ_{b}^{∗}(Ω_{L}) \ Trap(Ω_{L})

which defines a symplectic map on an open dense subset of Ṫ_{b}^{∗}(Ω_{K}) \ Trap(Ω_{K}), maps S_{b}^{∗}(Ω_{K}) \ Trap(Ω_{K}) onto S_{b}^{∗}(Ω_{L}) \ Trap(Ω_{L}), and such that F_{t}^{(L)} ∘ Φ = Φ ∘ F_{t}^{(K)} for all t ∈ IR and Φ = id on Ṫ_{b}^{∗}(IR^{n} \ U) \ Trap(Ω_{K}) = Ṫ_{b}^{∗}(IR^{n} \ U) \ Trap(Ω_{L}). Conversely, if K, L ∈ K_{0} are two obstacles for which there exists a homeomorphism Φ : S_{b}^{∗}(Ω_{K}) \ Trap(Ω_{K}) ⟶ S_{b}^{∗}(Ω_{L}) \ Trap(Ω_{L}) such that F_{t}^{(L)} ∘ Φ = Φ ∘ F_{t}^{(K)} for all t ∈ IR and Φ = id on S^{∗}(IR^{n} \ U) \ Trap(Ω_{K}), then K and L have the same SLS.

It is clear from Livshits’ example that the above conclusion is not true without any assumption about the sets of trapped points. In general it seems unlikely that SL_{K} provides substantial information about the size of the set Trap(Ω_{K}).

Denote by Kreg the set of those K ∈ K_{0} such that for almost all (x, ξ) ∈ S^{∗}(∂K) the set {t : F_{t}^{(K)}(x, ξ) ∈ S^{∗}(∂K)} is connected^{2}. One can show that Kreg is of second Baire category in K, i.e. the above property holds for generic obstacles.

Let U be a large ball containing the obstacles K and L. Given ξ ∈ S^{n−1} denote by Z_{ξ} the hyperplane in IR^{n} orthogonal to ξ and tangent to U such that U is contained in the open half-space R_{ξ} determined by Z_{ξ} and having ξ as an inner normal. Given an (ω, θ)-ray γ in Ω, the sojourn (travelling) time T_{γ} of γ is defined by T_{γ} = T′_{γ} − 2a, where T′_{γ} is the length of that part of γ which is contained in R_{ω} ∩ R_{−θ} and a is the radius of the ball U. It is known that this definition does not depend on the choice of the ball U.

3 Tangential singularities

Throughout this section K denotes an obstacle in IR^{n} such that the normal curvature of K does not vanish of infinite order. For brevity we will use the notation F_{t} = F_{t}^{(K)}. Let ϕ be a defining function for ∂K in a small neighbourhood V_{0} of ∂K. That is, φ : V_{0} ⟶ IR is smooth, dφ ≠ 0 on ∂K, and φ^{−1}(0) = ∂K. Define the Hamiltonian function p : T^{∗}(IR^{n}) ⟶ IR by p(x, ξ) = \frac{1}{2}(|ξ|^{2}−1). The corresponding Hamiltonian vector field is

H_{p} = (ξ_{1}, …, ξ_{n}; 0, …, 0) = ∑_{i=1}^{n} ξ_{i} \frac{∂}{∂x_{i}} .

As before, U will denote a fixed open ball in IR^{n} containing K. Set C = ∂U.

In this section we first study the sets

T_{k} = {σ ∈ S^{∗}(V_{0}) : H_{p}^{j}φ(σ) = 0 for j = 0, 1, …, k and H_{p}^{k+1}φ(σ) ≠ 0 },

where k is a positive integer. In general these sets are not manifolds, however locally each of them is contained in a submanifold of codimension 2 in S^{∗}(V_{0}). What is more important, it turns out that for k ≥ 2 the set T_{k} is locally contained in a submanifold of codimension 3 (see Proposition 3.1 below). An important consequence of this is that the set of those σ ∈ S^{∗}(IR^{n} \ U) that generate trajectories containing gliding segments on ∂K can be covered by a countable family of submanifolds of codimension 2 in S^{∗}(IR^{n} \ U), so topologically it does not divide S^{∗}(IR^{n} \ U).

Proposition 3.1. For each k ≥ 1 and each σ ∈ T_{k} there exists an open neighbourhood V (σ) of σ in T^{∗}(V_{0}) and a smooth submanifold Γ(σ) of V (σ) such that T_{k} ∩ V (σ) ⊂ Γ(σ) ⊂ S^{∗}(V_{0}) and the codimension of Γ(σ) in S^{∗}(V_{0}) is 3 for k = 1 and 4 for k ≥ 2. Consequently, as a submanifold of S^{∗}(V_{0}) the codimension of Γ(σ) is 2 for k = 1 and 3 for k ≥ 2.

2This means that the ray γ_{K}(σ) either has one single point of tangency to ∂K or it has one single gliding segment on ∂K and no other tangencies.

Proof. Denote by V′(σ) the set of those ρ ∈ T^{∗}(V_{0}) such that ξ ≠ 0 and H_{p}^{k+1}φ(ρ) ≠ 0, and let g : V′(σ) ⟶ IR be defined by g(ρ) = H_{p}φ(ρ). In coordinates we have

g(x, ξ) = ∑_{i=1}^{n} ξ_{i} \frac{∂φ}{∂x_{i}} (x) .

First, consider the case k = 1.We claim that

Γ = Γ(σ) = {ρ ∈ V′(σ) : p(ρ) = φ(ρ) = g(ρ) = 0}

is a submanifold of V (σ) = V′(σ). For this of course one has to show that dp(ρ), dφ(ρ) and dg(ρ) are linearly independent on Γ. Let ρ = (x, ξ) ∈ Γ and assume that

u dp(ρ) + a dφ(ρ) + b dg(ρ) = 0  (1)

for some u, a, b ∈ IR. Here d = d_{(x,ξ)}. Considering derivatives with respect to x_{j}, (1) implies

a \frac{∂φ}{∂x_{i}} (x) + b ∑_{i=1}^{n} ξ_{i} \frac{∂^{2}φ}{∂x_{i}∂x_{j}} (x) = 0.

Multiplying the latter by ξ_{j} and summing up gives

a H_{p}φ(ρ) + b H_{p}^{2}φ(ρ) = 0.

Since H_{p}φ(ρ) = 0 and H_{p}^{2}φ(ρ) ≠ 0, the above implies b = 0. Now considering derivatives with respect to ξ_{j} in (1), we get u = 0 and therefore a = 0, too. Hence Γ is a submanifold of V (σ) of codimension 3. Clearly, T_{1}∩V (σ) ⊂ Γ. Moreover S^{∗}(V_{0}) = p^{−1}(0)∩T^{∗}(V_{0}) shows that Γ ⊂ S^{∗}(V_{0}).

Next, consider the case k ≥ 2. Define another function f on V (σ) by

f(ρ) = H_{p}^{k}φ(ρ) = ∑_{i_{1},i_{2},...,i_{k}=1}^{n} ξ_{i_{1}}ξ_{i_{2}} … ξ_{i_{k}} \frac{∂^{k}φ}{∂x_{i_{1}}∂x_{i_{2}} … ∂x_{i_{k}}} (x) .

Set

Γ = {ρ ∈ V′(σ) : p(ρ) = φ(ρ) = g(ρ) = f(ρ) = 0}.

Clearly T_{k} ∩ V′(σ) ⊂ Γ ⊂ S^{∗}(V_{0}). We will show that dp, dφ, dg and df are linearly independent at any point ρ ∈ T_{k} ∩ V′(σ) (in particular at σ). Let ρ = (x, ξ) ∈ T_{k} ∩ V′(σ) and let

u dp(ρ) + a dφ(ρ) + b dg(ρ) + c df(ρ) = 0  (2)

for some u, a, b, c ∈ IR. Considering derivatives with respect to x_{m}, (2) implies

a \frac{∂φ}{∂x_{m}} (x) + b ∑_{i=1}^{n} ξ_{i} \frac{∂^{2}φ}{∂x_{i}∂x_{m}} (x) + c ∑_{i_{1},i_{2},...,i_{k}=1}^{n} ξ_{i_{1}}ξ_{i_{2}} … ξ_{i_{k}} \frac{∂^{k+1}φ}{∂x_{i_{1}}∂x_{i_{2}} … ∂x_{i_{k}}∂x_{m}} (x) = 0

for all m = 1, …, n. Multiplying the latter by ξ_{m} and summing up, we get

a H_{p}φ(ρ) + b H_{p}^{2}φ(ρ) + c H_{p}^{k+1}φ(ρ) = 0.

Since ρ ∈ T_{k} and k ≥ 2, we have H_{p}φ(ρ) = H_{p}^{2}φ(ρ) = 0 and H_{p}^{k+1}φ(ρ) ≠ 0. Hence c = 0. Next, considering the terms in (2) corresponding to derivatives with respect to ξ_{i}, we get

u ξ_{i} + b \frac{∂φ}{∂x_{i}} (x) = 0

for all i. Multiplying this by ξ_{i} and summing up gives 0 = u |ξ|^{2} + b H_{p}φ(ρ) = u |ξ|^{2}, so we must have u = 0 (since ξ ≠ 0 on V′(σ)). Returning to the previous equality and using the fact that dφ(ρ) = 0, one gets b = 0. Then (2) yields a = 0 as well. This shows in particular that dp, dφ, dg and df are linearly independent at σ. By continuity, there exists an open neighbourhood V (σ) of σ in V′(σ) such that dp, dφ, dg and df are linearly independent on V (σ). Then Γ(σ) = Γ′ ∩ V (σ) is a submanifold of codimension 4 in V (σ) and clearly T_{k} ∩ V (σ) ⊂ Γ(σ).

Next, we are going to show that for k ≥ 2, the set T_{k} can be covered by a countable family of codimension 2 submanifolds of S^{∗}(C).

Fix for a moment k ≥ 2 and m ≥ 0. Denote by T_{k}^{(m)} the set of those σ ∈ T_{k} such that there exists t > 0 with F_{t}(σ) ∈ S^{∗}(C) and the trajectory {F_{s}(σ) : 0 < s ≤ t} has no common points with ∪_{r=2}^{∞}T_{r} and has exactly m transversal reflection points at ∂K (and possibly some tangent points that belong to T_{1}). For σ ∈ T_{k}^{(m)} denote by s(σ) the minimal number t > 0 with F_{t}^{(K)}(σ) ∈ S^{∗}(C).

Lemma 3.2. For every σ_{0} ∈ T_{k}^{(m)} there exists an open neighbourhood U_{k}^{(m)}(σ_{0}) such that the set

N_{k}^{(m)}(σ_{0}) = {F_{s(σ)}(σ) : σ ∈ T_{k}^{(m)} ∩ U_{k}^{(m)}(σ)}

is contained in a smooth codimension 2 submanifold of S^{∗}(C).

Proof. We will use the above proposition and an argument from [St2].

Fix an arbitrary σ_{0} ∈ T_{k}^{(m)} and let s_{1} < s_{2} < … < s_{m} be the times of the transversal reflections of {F_{s}(σ_{0}) : 0 < s ≤ s(σ_{0})}. Clearly 0 < s_{1} and s_{m} < s(σ). For each j = 1, …, m fix two numbers a_{j} and b_{j} close to s_{j} and such that

b_{0} = 0 < a_{1} < s_{1} < b_{1} < a_{2} < s_{2} < b_{2} < … < a_{m} < s_{m} < b_{m} < a_{m+1} = s(σ_{0}).

For each j = 1, …, m choose arbitrary smooth cros-section (i.e. submanifolds of S^{∗}(Ω) of codimension 1) A_{j} and B_{j} to the trajectory {F_{s}(σ_{0}) : 0 < s ≤ s(σ_{0})} such that F_{a_{j}}(σ_{0}) ∈ A_{j} and F_{b_{j}}(σ_{0}) ∈ B_{j}. We assume that A_{j} and B_{j} are so small and so close to the reflection point F_{s_{j}}(σ_{0}) that for any ρ ∈ A_{j} the trajectory of ρ under F_{t} makes exactly one (transversal) reflection at ∂K before intersecting transversally B_{j}.

Let V (σ_{0}) be an open neighbourhood of σ_{0} in S^{∗}(V_{0}) with the properties described in Proposition 3.1. For ρ in a small neighbourhood U of σ_{0} in V (σ_{0}) we denote by ψ_{t}(ρ) the unique curve in S^{∗}(IR^{n} for which there exists a sequence of numbers

b_{0}(ρ) = 0 < a_{1}(ρ) < s_{1} < b_{1}(ρ) < a_{2}(ρ) < … < a_{m}(ρ) < s_{m} < b_{m}(ρ) < a_{m+1}(ρ)

with the following properties:

(i) ψ_{a_{j}(ρ)}(ρ) ∈ A_{j} and ψ_{b_{j}(ρ)}(ρ) ∈ B_{j} for all j = 1, …, m, ψ_{a_{m+1}(ρ)}(ρ) ∈ S^{∗}(C), and ψ_{s}(ρ) ∈ S^{∗}(U) for s ∈ (0, a_{m+1}(ρ));

(ii) for each j = 0, 1, …, m, the curve {ψ_{t}(ρ) : b_{j}(ρ) ≤ t ≤ a_{j+1}(ρ)} is a trajectory of the vector field H_{p} in IR^{n};

(iii) for each j = 1, …, m, the curve {ψ_{t}(ρ) : a_{j}(ρ) ≤ t ≤ b_{j}(ρ)} is a trajectory of the GHF F_{t} in Ω.

It is clear that if the neighbourhood U of σ_{0} in V (σ_{0}) is sufficiently small, then the curve ψ_{t}(ρ) is well-defined for all ρ ∈ U.

Set Λ(σ_{0}) = {ρ ∈ U : H_{p}^{k}φ(ρ) = 0 }. We assume that on V (σ_{0}) we have H_{p}^{k+1}φ ≠ 0 (which follows from the construction of V (σ_{0}) in the proof of Proposition 3.1). Then Λ(σ_{0}) is a codimension 1 submanifold of U transversal to the vector field H_{p}. Consequently, the map Λ(σ_{0}) ∋ ρ ↦ ψ_{a_{1}(ρ)}(ρ) ∈ A_{1} is smooth and a local bijection, so it defines a local diffeomorphism. Dealing in the same way with the shift along the curve ψ_{t}(ρ) between successive cros-sections, one derives that the map Ψ_{k}^{(m)}(ρ) = ψ_{a_{m+1}(ρ)}(ρ) from U to S^{∗}(C) is a local diffeomorphism.

Set U_{k}^{(m)}(σ_{0}) = U. It follows from Proposition 3.1 that Γ(σ_{0}) ∩ U_{k}^{(m)} is a codimension 2 submanifold of Λ(σ_{0}). Hence Ψ_{k}^{(m)}(Γ(σ_{0}) ∩ U_{k}^{(m)}(σ_{0})) is a a codimension 2 submanifold of S^{∗}(C). It remains to show that N_{k}^{(m)}(σ_{0}) = Ψ_{k}^{(m)}(Γ(σ_{0}) ∩ U_{k}^{(m)}(σ_{0})). To check this, observe that for any ρ ∈ U_{k}^{(m)}(σ_{0})∩T_{k}^{(m)} we have ψ_{s}(ρ) = F_{s}^{(K)}(ρ) for all s ∈ [0, a_{m+1}(ρ)]. In particular s(σ) = a_{m+1}(ρ) and therefore N_{k}^{(m)}(σ_{0}) = Ψ_{k}^{(m)}(Γ(σ_{0}) ∩ U_{k}^{(m)}(σ_{0})). This proves the lemma.

Denote by G_{K} the set of those σ ∈ S^{∗}(C) such that F_{t}(σ) ∈ T_{k} for some t ∈ IR and some k ≥ 2. Clearly G_{K} contains any σ ∈ S^{∗}(C) that generates a trajectory containing a gliding segment on ∂K.

As an immediate consequence of the above lemma we get the following.

Proposition 3.3. There exists a countable family {N_{i}} of codimension 2 submanifolds of S^{∗}(C) such that ∪_{i}N_{i} contains all points σ ∈ S^{∗}(C) such that F_{t}^{(K)}(σ) ∈ T_{k} for some k ≥ 2.

Proof. Cover the set T_{k}^{(m)} with a countable family U_{k}^{(m)}(σ_{j}^{(k,m)}) of open subsets of S^{∗}(V_{0}) with the properties listed in Lemma 3.2. Then

G_{K} ⊂ ∪_{j=1}^{∞} ∪_{m=0}^{∞} ∪_{k=1}^{∞}N_{k}^{(m)}(σ_{j}^{(k,m)}) ,

so the statement follows from Lemma 3.2.

Finally, it remains to deal with the case k = 1.

Lemma 3.4. For every σ_{0} ∈ T_{1} there exists an open neighbourhood U(σ) such that the set

N(σ_{0}) = {F_{s(σ)}(σ) : σ ∈ T_{1} ∩ U(σ_{0})}

is contained in a smooth codimension 1 submanifold of S^{∗}(C).

Proof. This is essentially a repetition of the proof of Lemma 3.2 with minor modifications and less technicalities, since there are no futher tangencies that have to be avoided. We omit the details.

An important consequence of the last lemma is the following.

Proposition 3.5. There exists a countable family {M_{i}} of codimension 1 submanifolds of S^{∗}(C) \ (Trap(Ω_{K}) ∪ G_{K}) such that every σ ∈ S^{∗}(C) \ (Trap(Ω_{K}) ∪ G_{K} ∪ ∪_{i}M_{i}) generates a simply reflecting trajectory in Ω. Moreover the family {M_{i}} is locally finite in S^{∗}(C) \ (Trap(Ω_{K}) ∪ G_{K}), that is any compact subset of S^{∗}(C)\(Trap(Ω_{K})∪G_{K}) has common points with only finitely many of the submanifolds M_{i}.

Proof. For every σ ∈ T_{1} \ Trap(Ω_{K}) choose an open neighbourhood U(σ) as in Lemma 3.4. Shrinking U(σ) if necessary, we may assume that H_{p}^{2}φ ≠ 0 on the closure of U(σ) and therefore it does not have common points with T_{k} for any k ≥ 2. Also, we choose U(σ) such that U(σ) ∩ Trap(Ω_{K}) = ∅. Consequently, dist(N(σ), Trap(Ω_{K}) ∪ G_{K}) > 0, and so there exists a smooth codimension 1 submanifold M(σ) of S^{∗}(C) such that N(σ) ⊂ M(σ) and the closure of M(σ) has no common points with Trap(Ω_{K}) ∪ G_{K}.

Choose a countable set of elements σ_{i} ∈ T_{1} \ Trap(Ω_{K}) such that T_{1} \ Trap(Ω_{K}) ⊂ ∪_{i}U(σ_{i}) and {U(σ_{i}) ∩ T_{1}} is a locally finite family in T_{1} \ Trap(Ω_{K}). Denote M_{i} = M(σ_{i}). It is now clear that any σ ∈ S^{∗}(C) \ (Trap(Ω_{K}) ∪ G_{K} ∪ ∪_{i}M_{i}) generates a simply reflecting trajectory in Ω. It remains to show that {M_{i}} is locally finite in S^{∗}(C) \ (Trap(Ω_{K}) ∪ G_{K}). Let L be a compact subset of S^{∗}(C) \ (Trap(Ω_{K}) ∪ G_{K}). Then the sojourn (travelling) times of trajectories generated by elements of L are uniformly bounded. Assume that there exists ρ_{m} ∈ M_{i_{m}} ∩ L for infinitely many i_{m}. Choosing a subsequence, we may assume that ρ_{m} → ρ ∈ L as m → ∞. Since ρ_{m} ∈ M_{i_{m}}, there exists t_{m} ∈ IR with F_{t_{m}}(ρ_{m}) ∈ U(σ_{i_{m}}) ∩ T_{1}. Since the sequence {t_{m}} is bounded, we may assume that t_{m} → t ∈ IR as t → ∞. Then F_{t_{m}}(ρ_{m}) → F_{t}(ρ), so for σ = F_{t}(ρ) we must have H_{p}φ(σ) = 0. On the other hand, ρ ∈ L shows that ρ ∉ Trap(Ω_{K}) ∪ G_{K}, so the trajectory generated by ρ is not trapped and does not have common points with T_{k} for any k ≥ 2. Hence σ ∈ T_{1} \ Trap(Ω_{K}), and therefore there exists a compact neighbourhood W of σ in S^{∗}(IR^{n}) such that W ∩ T_{1} ∩ Trap(Ω_{K}) = ∅. Then F_{t_{m}}(ρ_{m}) → σ implies that W has common points with U(σ_{i_{m}}) ∩ T_{1} for all sufficiently large m, contradiction with the local finiteness of the family {U(σ_{i})∩T_{1}} in T_{1}\Trap(Ω_{K}). Hence L can have common points with only finitely many M_{i}’s.

4 Proof of Theorem 1.1

Assume that K ∈ Kreg and L ∈ Kreg are two obstacles in IR^{n} with real analytic boundaries that have almost the same SLS and such that every connected component of K (resp. L) has at least one common point with a scattering ray in Ω_{K} (resp. Ω_{L}). As before, U will denote a fixed open ball containing K and L and C will be the boundary sphere of U.

For K we assume in addition that the set Trap(Ω_{K}) is not topologically dividing S^{∗}(C), that is, S^{∗}(C) \ Trap(Ω_{K}) is connected. Then L has the same property, since K and L having almost the same SLS and Theorem 2.1 imply Trap(Ω_{L}) = Trap(Ω_{K}).

Under these assumptions we will show that L = K.

Clearly some connected components of K coincide with connected components of L. Indeed, it follows from result of Majda [Ma] and Lax and Phillips [LP2] (and also from the more general Theorem 2.1 above) that the extremal points of K and L coincide. The real analyticity of ∂K and ∂L then implies that any connected component of K containing extremal points of K coincides with a corresponding connected component of L.

Fix a countable family {N_{i}} of codimension 2 submanifolds of S^{∗}(C) with the properties described in Proposition 3.3 and a countable family {M_{i}} of codimension 1 submanifolds of S^{∗}(C) \ (Trap(Ω_{K}) ∪ G_{K}) with the properties described in Proposition 3.5.

Denote by X the union of all connected components of ∂K^{(ob)} contained in ∂L. Then Y = ∂K^{(ob)} \ X is a finite union of connected components of ∂K, so Y (if not empty) is a compact real analytic submanifold of IR^{n} and X ∩ Y = ∅. Our aim is to show that Y = ∅.

Assume that Y ≠ ∅. It then follows from the assumptions about K that there exists σ′ ∈ S^{∗}(C) such that the trajectory

γ_{K}(σ′) = {pr_{1}(F_{t}^{(K)}(σ′)) : t ∈ IR}

has at least one common point with Y. Since both G_{K} ∪ ∪_{i}M_{i} and Trap(Ω_{K}) ∩ S^{∗}(C) have Lebesgue measure zero in S^{∗}(C) (cf. [St2]), slightly changing σ′, we may assume that γ_{K}(σ′) is a simply reflecting ray having a common point with Y (which is therefore a transversal reflection point). Clearly, there exists points σ ∈ S^{∗}(C) such that γ_{K}(σ) has no common points with ∂K. Fix an arbitrary σ_{0} ∈ S^{∗}(C) with this property.

Since σ_{0}, σ′ ∈ S^{∗}(C) \ Trap(Ω_{K}) and this set is open and connected (by assumption), there exists a smooth curve σ(s), 0 ≤ s ≤ 1, in S^{∗}(C) \ Trap(Ω_{K}) such that σ(0) = σ_{0}, σ(1) = σ′. Since Trap(Ω_{K}) ∩ S^{∗}(C) is a compact set, there exists a whole compact neighbourhood W of γ = {σ(s) : s ∈ [0, 1]} in S^{∗}(C) such that W ∩ Trap(Ω_{K}) = ∅. Slightly perturbing the curve γ in the interior of W, we may assume that γ is transversal to any of the submanifolds M_{i} and N_{i} (cf. [Hi]). Now the fact that the codimension of each M_{i} in S^{∗}(C) is 2 implies that γ ∩M_{i} = ∅ for each i. Thus, γ ∩ G_{K} = ∅. On the other hand, the set G_{K} ∩ W is closed in W (since W is away from Trap(Ω_{K})), and so G_{K} ∩ W is a compact subset of W having no common points with γ. Then shrinking W, we may assume that W ∩G_{K} = ∅. With this, we have W ∩(G_{K} ∪Trap(Ω_{K})) = ∅. Now the fact that the family {M_{i}} is locally finite in S^{∗}(C) \ (Trap(Ω_{K}) ∪ G_{K}) implies that γ ∩ M_{i} ≠ ∅ only for finitely many i. Consequently, there exist only finitely many s_{1}, …, s_{m} ∈ [0, 1] such that γ_{K}(σ(s_{j})) contains a tangent point to ∂K (and that must be a point from T_{1}). In general it may happen that some of the trajectories γ_{K}(σ(s_{j})) have more than one tangency to ∂K. However, using the condition K ∈ Kreg, we can slightly perturb the curve γ near each s_{j}, so that γ_{K}(σ(s)) has a single point of tangency to ∂K for any s ∈ [0, 1] with σ(s) ∈ ∪_{i}M_{i}.

In this way, we proved the existence of a smooth curve σ(s) (s ∈ [0, 1]) in S^{∗}(C) \ (G_{K} ∪ Trap(Ω_{K})) such that σ(0) = σ_{0}, σ(1) = s′, there exist only finitely many s ∈ [0, 1] with σ(s) ∈ ∪_{i}M_{i} and for any such s the trajectory γ_{K}(σ(s)) has at most one tangent point to ∂K.

Since γ ∩ Trap(Ω_{K}) = ∅ and Y is compact, there exists

a = min{s ∈ [0, 1] : γ_{K}(σ(s)) ∩ Y ≠ ∅}.

Clearly a > 0 and for ρ = σ(a) we have γ_{K}(ρ) ∩ Y ≠ ∅. The choice of a shows that every common point of γ_{K}(ρ) with Y must be a tangent point to Y. So, a ∈ M_{i} for some i. Consequently, γ_{K}(ρ) has only one tangent point y to Y, and this is its only common point with Y. Now we no longer nead the curve γ. There exists a point ρ′ ∈ W close to ρ such that γ_{K}(ρ′) is a simply reflecting ray in Ω_{K} and has exactly one common point y′ with Y, y′ close to y.

We claim that y′ and all points of Y close to y′ belong to ∂L. Indeed, let y_{1}, …, y_{j−1}, y_{j} = y′, …, y_{m} be the successive reflection points of γ(ρ′). By the construction of ρ′, we have y_{i} ∈ X for all i ≠ j. The definition of X then implies y_{i} ∈ ∂K ∩∂L and the tangent planes to ∂K and ∂L at y_{i} coincide for all i ≠ j. From this and Theorem 2.1 above one concludes that γ_{K}(ρ′) = γ_{L}(ρ′), so we must have y′ ∈ ∂L. In the same way, considering points ρ′′ ∈ W sufficiently close to ρ′, one shows that all y′′ ∈ Y sufficiently close to y′ belong to ∂L. Now the real analyticity of ∂K and ∂L implies that y′ belongs to a connected component of ∂K that is contained in ∂L, contradiction with the definition of the sets X and Y. Hence we must have Y = ∅, i.e. ∂K^{(ob)} ⊂ ∂L. By symmetry, ∂L^{(ob)} ⊂ ∂K, so ∂K^{(ob)} = ∂L^{(ob)}.

5 The case of several strictly convex bodies

In this section we assume that K is an obstacle in IR^{n} (n ≥ 3) of the form

K = K_{1} ∪ K_{2} ∪ … ∪ K_{s} ,  (3)

where K_{i} are compact strictly convex disjoint domains in IR^{n} with C^{∞} boundaries ∂K_{i}. Set

d = min_{i≠j} dist(K_{i}, K_{j}),

and F = F^{(K)}.

Lemma 5.1. Let X be a C^{∞} smooth submanifold of codimension 1 in IR^{n} , and let x_{0} ∈ X and ξ_{0} ∈ T_{x}X, ∥ξ_{0}∥ = 1, be such that the normal curvature of X at x_{0} in the direction ξ_{0} is non-zero. Then for every ϵ > 0 there exist an open neighbourhood V of x_{0} in X, a smooth map V ∋ x ↦ ξ(x) ∈ T_{x}X and a smooth positive function t(x) ∈ [δ, ϵ] on V for some δ > 0 such that Y = {y(x) = x + t(x)ξ(x) : x ∈ V } is a smooth strictly convex surface with unit normal field μ(y(x)) = ξ(x), x ∈ V. That is, the normal field of Y consists of vectors tangent to X at the corresponding points of V.

Proof. Considering X with the Riemann metric induced by IR^{n}, there exists a local smooth codimension 1 submanifold X′ of X containing x and perpendicular to ξ_{0} at x_{0} and such that the second fundamental form of X′ in X with respect to the normal ξ_{0} is negative definite at x_{0}. (For example, we can take another strictly convex (n − 1)-dimensional submanifold Z of IR^{n} containing x_{0} and having ”outward” unit normal ξ_{0} at x_{0}, and set X′ = X ∩ Z.) Parametrize X′ by h(u′), u′ = (u_{2}, …, u_{n−1}), h(0) = x_{0}, and let ξ(u′) be a continuous unit normal field to X′ with ξ(u′) ∈ T_{h(u′)}X for all u′ and ξ(0) = ξ_{0}. For any u′ let c(t; u′) be the geodesic on X parametrized by arc length t such that c(0; u′) = h(u′) and ċ(t; u′) = ξ(u′). Define r(u_{1}, u_{2}, …, u_{n−1}) = c(u_{1}; u′). It then follows that for |u_{1}| and ∥u′∥ small enough, r(u) is a smooth parametrization of an open neighbourhood V of x_{0} in X such that x_{0} = r(0), ξ_{0} = \frac{∂r}{∂u_{1}} (0), and

∥\frac{∂r}{∂u_{1}} (u)∥ = 1 , ⟨\frac{∂r}{∂u_{1}} (u), \frac{∂r}{∂u_{i}} (u)⟩ = 0 , i > 1 , u ∈ U .  (4)

(cf. e.g. Theorems 3 and 4 in Sect. 8.1 of [BC] for the second relation). Shrinking the neighbourhood V of x_{0} if necessary, we may assume that u runs over some open ball U in IR^{n−1}. Notice that the second fundamental form of X′ in X at x_{0} has the form

II′(v′) = ∑_{i,j=2}^{n−1} v_{i}v_{j} ⟨ξ_{0}, \frac{∂^{2}r}{∂u_{i}∂u_{j}} (0)⟩ ,

where v′ = (v_{2}, …, v_{n−1}) ∈ IR^{n−2}. So the choice of X′ implies II′(v′) < 0 whenever v′ ≠ 0.

Fix a small ϵ > 0 and set y(u) = r(u) + (ϵ − u_{1})\frac{∂r}{∂u_{1}} (u), u ∈ U. Shrinking the ball U if necessary, we may assume that |u_{1}| < ϵ/2 for all u ∈ U. It will become clear later how small ϵ should be.

We claim that Y = {y(u) : u ∈ U} is a smooth submanifold of IR^{n} (provided ϵ > 0 and U are small enough), y(u) is a smooth parametrization of Y and μ(u) = \frac{∂r}{∂u_{1}} (u) is a normal vector to Y at y(u).

First, notice that differentiating (3) with respect to u_{j} implies

⟨\frac{∂r}{∂u_{1}} (u), \frac{∂^{2}r}{∂u_{1}∂u_{j}} (u)⟩ = 0 , 1 ≤ j ≤ n − 1 ,  (5)

⟨\frac{∂^{2}r}{∂u_{1}∂u_{j}} (u), \frac{∂r}{∂u_{i}} (u)⟩ + ⟨\frac{∂r}{∂u_{1}} (u), \frac{∂^{2}r}{∂u_{i}∂u_{j}} (u)⟩ = 0 , 2 ≤ i ≤ n − 1 , 1 ≤ j ≤ n − 1 . (6)

From these two equalities, it follows in particular that \frac{∂^{2}r}{∂u_{1}^{2}} (u) ⊥ \frac{∂r}{∂u_{i}} (u) for any i = 1, …, n − 1,

and therefore \frac{∂^{2}r}{∂u_{1}^{2}} (u) is a normal vector to X at r(u). On the other hand, the assumption that the normal curvature of X at x_{0} = r(0) in the direction of ξ_{0} = \frac{∂r}{∂u_{1}} (0) is non-zero implies that \frac{∂^{2}r}{∂u_{1}^{2}} (0) ≠ 0. In particular, the vectors \frac{∂^{2}r}{∂u_{1}^{2}} (0), \frac{∂r}{∂u_{2}} (0), …, \frac{∂r}{∂u_{n−1}} (0) in IR^{n} are linearly independent. Without loss of generality we may assume that the matrix formed by the first n − 1 coordinates of these vectors has a non-zero determinant.

Since

\frac{∂y}{∂u_{1}} (u) = (ϵ − u_{1})\frac{∂^{2}r}{∂u_{1}^{2}} (u) , \frac{∂y}{∂u_{i}} (u) = \frac{∂r}{∂u_{i}} (u) + (ϵ − u_{1})\frac{∂^{2}r}{∂u_{1}∂u_{i}} (u) , i > 1,  (7)

(4) and (5) imply that μ(u) = \frac{∂r}{∂u_{1}} (u) is a normal vector to Y at y(u), provided Y is a smooth submanifold with parametrization y(u).

To prove that Y is locally a smooth (n − 1)-dimensional submanifold of IR^{n}, it is enough to show that if ϵ > 0 is small enough, then the vectors \frac{∂y}{∂u_{i}} (0) (1 ≤ i ≤ n − 1) are linearly independent. Assume the contrary, i.e. for any ϵ ∈ (0, a] for some a > 0, the corresponding vectors \frac{∂y}{∂u_{i}} (0) are linearly dependent. Let y(u) = (y_{1}(u), …, y_{n}(u)) and r(u) = (r_{1}(u), …, r_{n}(u)). Define z(u) = (y_{1}(u), …, y_{n−1}(u)) and h(u) = (r_{1}(u), …, r_{n−1}(u)). Then the vectors \frac{∂z}{∂u_{i}} (0) (1 ≤ i ≤ n − 1) are also linearly dependent, so we must have

0 = det \frac{∂z}{∂u}(0) = det ( \begin{matrix}\frac{∂z}{∂u_{1}} (0) \\ \frac{∂z}{∂u_{2}} (0) \\ … \\ \frac{∂z}{∂u_{n−1}} (0) \end{matrix} ) = det ( \begin{matrix}ϵ\frac{∂^{2}h}{∂u_{1}^{2}}(0) \\ \frac{∂h}{∂u_{2}} (0) + ϵ\frac{∂^{2}h}{∂u_{1}∂u_{2}} (0) \\ … & … \\ \frac{∂h}{∂u_{n−1}} (0) + ϵ\frac{∂^{2}h}{∂u_{1}∂u_{n−1}} (0) \end{matrix} ) ,

where the rows in the matrices above are vectors in IR^{n−1}. Dividing by ϵ the first row of the determinant in the right-hand side and then letting ϵ → 0, we obtain that the matrix formed by the first n − 1 coordinates of the vectors \frac{∂^{2}r}{∂u_{1}^{2}} (0), \frac{∂r}{∂u_{2}} (0), …, \frac{∂r}{∂u_{n−1}} (0) has a zero determinant – contradiction with our assumption above.

Thus, for ϵ > 0 sufficiently small the vectors \frac{∂y}{∂u_{i}} (0) (1 ≤ i ≤ n − 1) are linearly independent. Shrinking U if necessary, we may assume that \frac{∂y}{∂u_{i}} (u) (1 ≤ i ≤ n − 1) are linearly independent for any u ∈ U. Then Y is a smooth (n − 1)-dimensional submanifold of IR^{n} and y(u) is a smooth parametrization of Y. As we observed above, μ(u) = \frac{∂r}{∂u_{1}} (u) is then a unit normal to Y at y(u). Moreover, by the definition of y(u), the segment [r(u), y(u)] is tangent to X at r(u).

It remains to show that the normal curvature of Y with respect to the normal field μ(u) is negative. For this we need the second derivatives of y(u) which we get from (7):

\frac{∂^{2}y}{∂u_{1}^{2}} (0) = −\frac{∂^{2}r}{∂u_{1}^{2}} (0) + ϵ\frac{∂^{3}r}{∂u_{1}^{3}} (0) ,

\frac{∂^{2}y}{∂u_{1}∂u_{i}} (0) = ϵ\frac{∂^{3}r}{∂u_{1}^{2}∂u_{i}} (0) , 2 ≤ i ≤ n − 1 ,

\frac{∂^{2}y}{∂u_{i}∂u_{j}} (0) = \frac{∂^{2}r}{∂u_{i}∂u_{j}} (0) + ϵ\frac{∂^{3}r}{∂u_{1}∂u_{i}∂u_{j}} (0) , 2 ≤ i, j ≤ n − 1 .

Hence the coefficients c_{ij} = ⟨μ(0), \frac{∂^{2}y}{∂u_{i}∂u_{j}} (0)⟩ of the second fundamental form of Y at y(0) have the form:

c_{1j} = ϵ ⟨\frac{∂r}{∂u_{1}} (0), \frac{∂^{3}r}{∂u_{1}∂u_{1}^{2}∂u_{j}} (0)⟩ , 2 ≤ j ≤ n − 1 ,

c_{ij} = ⟨\frac{∂r}{∂u_{1}} (0), \frac{∂^{2}r}{∂u_{i}∂u_{j}} (0)⟩ + ϵ ⟨\frac{∂r}{∂u_{1}} (0), \frac{∂^{3}r}{∂u_{1}∂u_{i}∂u_{j}} (0)⟩ , 2 ≤ i, j ≤ n − 1 .

On the other hand, differentiating (5) with respect to u_{i}, one gets

⟨\frac{∂r}{∂u_{1}} (0), \frac{∂^{3}r}{∂u_{1}∂u_{i}∂u_{j}} (0)⟩ = − ⟨\frac{∂^{2}r}{∂u_{1}∂u_{i}} (0), \frac{∂^{2}r}{∂u_{1}∂u_{j}} (0)⟩ .

Thus, the second fundamental form II(v) (v = (v_{1}, …, v_{n−1}) ∈ IR^{n−1}) of Y at y(0) has the form

II(v) = ∑_{i,j=1}^{n−1} v_{i}v_{j}c_{ij} = ∑_{i,j=1}^{n−1} v_{i}v_{j} ⟨\frac{∂r}{∂u_{1}} (0), \frac{∂^{3}r}{∂u_{1}∂u_{i}∂u_{j}} (0)⟩ + ∑_{i,j=2}^{n−1} v_{i}v_{j} ⟨\frac{∂r}{∂u_{1}} (0), \frac{∂^{2}r}{∂u_{i}∂u_{j}} (0)⟩

= − ∑_{i,j=1}^{n−1} v_{i}v_{j} ⟨\frac{∂^{2}r}{∂u_{1}∂u_{i}} (0), \frac{∂^{2}r}{∂u_{1}∂u_{j}} (0)⟩ + II′(v′) = −∥w∥^{2} + II′(v′),

where v′ = (v_{2}, …, v_{n−1}) and w = ∑_{i=1}^{n−1} v_{i} \frac{∂^{2}r}{∂u_{1}∂u_{i}} (0). Since by the choice of the submanifold X′ we have II′(v′) < 0 for v′ ≠ 0, it now follows that II(v) < 0 whenever v ≠ 0. Thus, Y is strictly convex at y(0) with respect to the normal μ(0) = \frac{∂r}{∂u_{1}} (0), and therefore the same conclusion holds for any y(u) close enough to y(0). So, shrinking U (and therefore V) we get that the whole submanifold Y is strictly convex in IR^{n}.

Let σ_{0} = (x_{0}, ξ_{0}) ∈ S^{∗}(∂K) be such that ρ_{0} = F_{t_{0}}(σ_{0}) ∈ S^{∗}(∂K) for some t_{0} > 0 and t_{0} is minimal with this property. Let φ(x) and p(ξ) be as in the beginning of Sect. 3. On some small neighbourhoods U_{0} and W_{0} of σ_{0} and ρ_{0} in T^{∗}(IR^{n}) define

M = {σ ∈ U_{0} : p(σ) = H_{p}φ(σ) = 0} , N = {σ ∈ W_{0} : p(σ) = H_{p}φ(σ) = 0},

M_{0} = M ∩ φ^{−1}(0), N_{0} = N ∩ φ^{−1}(0). It follows from Sect. 3 that M and M_{0} (resp. N and N_{0}) are smooth submanifolds of U_{0} (resp. W_{0}) of codimensions 2 and 3. The strict convexity of ∂K implies that the Poisson bracket {p, H_{p}φ} = H_{p}^{2}φ ≠ 0 on each of the sets M and N. Therefore M and N are codimension 2 symplectic submanifolds of U_{0} and W_{0}, respectively. Moreover, H_{φ}p = 0

and H_{φ}(H_{p}φ)(x, ξ) = ∑_{i=1}^{n} (\frac{∂φ}{∂x_{i}} )^{2} > 0 imply that M_{0} and N_{0} are codimension 1 submanifolds of M and N, respectively.

Let t_{1} < … < t_{m} be the times of reflections of the trajectory γ = {F_{t}(σ_{0}) : 0 < t < t_{0}}. Take a small open neighbourhood U_{0} of σ_{0} in T^{∗}(IR^{n}) and for each σ ∈ U_{0} define the curve ψ_{t}(σ), 0 ≤ t ≤ t_{0} + d/2, as the billiard trajectory of σ in IR^{n} which makes (transversal) reflections at ∂K near the points F_{t_{i}}(σ_{0}) (i = 1, …, m) but disregards ∂K near σ_{0} and ρ_{0}. Let Φ : M′ ⟶ N′ be the map defined by shift along the curves ψ_{t}, that is for any σ ∈ M′, Φ(σ) = ψ_{t}(σ), where t is such that |t − t_{0}| < d/2 and ψ_{t}(σ) ∈ N′. If U_{0} and V_{0} are small enough, then Φ is well-defined and smooth and Φ(M) ⊂ N. Moreover, the restriction of Φ ot M determines a symplectic map Φ : M ⟶ N.

Lemma 5.2. Under the assumptions and with the notation above, if the open neighbourhoods U_{0} and W_{0} of σ_{0} and ρ_{0} in T^{∗}(IR^{n}) are sufficiently small, then M_{0} ∩ Φ^{−1}(N_{0}) is a submanifold of M_{0} of positive codimension.

Proof. We will need the submanifolds

M′ = {σ ∈ U_{0} : H_{p}φ(σ) = 0} , N′ = {σ ∈ W_{0} : H_{p}φ(σ) = 0},

of codimension 1 in U_{0} and W_{0}, respectively.

It follows from Lemma 5.1 that there exists an open neighbourhood V of x_{0} in ∂K, a smooth map V ∋ x ↦ ξ(x) ∈ T_{x}∂K and a smooth positive function t(x) ∈ [δ, ϵ] on V for some ϵ > δ > 0 such that Y = {y(x) = x + t(x)ξ(x) : x ∈ V } is a smooth strictly convex surface with unit normal field μ(y(x)) = ξ(x), x ∈ V. Taking V sufficiently small, we may assume that Y is entirely in the exterior of K. Then Ỹ ⊂ p^{−1}(0) and Ỹ is an (n − 1)-dimensional submanifold of T^{∗}(IR^{n}). It is contained in the codimension 1 submanifold L′ = T_{Y}^{∗} (IR^{n}) of T^{∗}(IR^{n}). Clearly, (y_{0}, η_{0}) = (y(x_{0}), μ(x_{0})) = F_{s_{0}}(σ_{0}) for s_{0} = t(x_{0}).

Define the local maps Ξ : M′ ⟶ L′ and Ψ : L′ ⟶ N′ as the shifts along the curves ψ_{t}; then Φ = Ψ ∘ Ξ. Notice that

Ξ^{−1}(Ỹ) = {(x, ξ(x)) : x ∈ V } ⊂ M_{0} ⊂ M

is an (n − 1)-dimensional (Lagrangian) submanifold of M.

Fix an arbitrary s > 0 such that t_{m} < s + s_{0} < t_{0} and consider the set

S = {pr_{1}(ψ_{s}(ρ)) : ρ ∈ Ỹ }.

Then

ψ_{s}(Ỹ) = {(q, ω(q)) : q ∈ S},

where ω(q) is the unit normal field on Z pointing in the direction of the flow F on ψ_{s}(Ỹ). Clearly S is the shift of the strictly convex surface Y along the billiard flow in the exterior of K after fixed time s. It is known in the theory of dispersing billiards (see Sinai [Si1], [Si2]) that the strict convexity of ∂K and that of Y implies that S is a smooth (n − 1)-dimensional submanifold of IR^{n} which is strictly convex with respect to the normal field determined by the direction of the flow. Assuming that the neighbourhood V of x_{0} in ∂K, and therefore Y and S, are sufficiently small, we have ψ_{t}(q, ω(q)) = (q + tω(q), ω(q)) for 0 ≤ t < t_{0} + d/2 − (s + s_{0}). It is now easy to see that the set S_{0} of those q ∈ S such that ψ_{t}(q, ω(q)) ∈ S^{∗}(∂K) for some t ∈ (0, t_{0} + d/2 − (s + s_{0})) is a smooth submanifold of S of codimension 1 (cf. Lemma 5.3 below for a proof of this fact). Then {(q, ω(q)) : q ∈ S_{0}} has the same property in ψ_{s}(Ỹ), and taking (ψ_{s})^{−1}, we deduce that the set L_{0} of those ρ ∈ Ỹ such that F_{t}(ρ) ∈ S^{∗}(∂K) for some t with |t − t_{0} − s_{0}| < d/2 is a smooth codimension 1 submanifold of Ỹ.

Assume that U_{0} is so small that pr_{1}(U_{0}) ⊂ V. Notice that L_{0} = Ỹ ∩Ψ^{−1}(N_{0}). Applying Ξ^{−1} to this relation we get that Ξ^{−1}(L_{0}) = Ξ^{−1}(Ỹ) ∩ Φ^{−1}(N_{0}). Here L = Ξ^{−1}(Ỹ) is a submanifold of M_{0} of dimension n − 1, while Ξ^{−1}(L_{0}) is a submanifold of M_{0} of dimension n − 2. Since L ∩ Φ^{−1}(N_{0}) is a proper submanifold of L and L ⊂ M_{0}, it follows that M_{0} ∩ Φ^{−1}(N_{0}) is a proper submanifold of M_{0}.

Given a smooth (n − 1)-dimensional submanifold X of IR^{n} with a specified continuous unit normal field ν_{X}, we will denote

X̃ = {(x, ν_{X}(x)) : x ∈ X} ⊂ S_{X}^{∗}(IR^{n}).

The following simple lemma is well-known. We prove it for completeness (it was already used in the proof of Lemma 5.2 above).

Lemma 5.3. Let Z and X be smooth (n − 1)-dimensional submanifold of IR^{n} with specified continuous unit normal fields ν_{Z} and ν_{X} such that Z is convex with respect to ν_{Z} while X is strictly convex with respect to ν_{Z}. Let z_{0} ∈ Z be such that x_{0} = z_{0} + τ_{0}ν_{Z}(z_{0}) ∈ X for some τ_{0} > 0 and the ray θ(z_{0}) = {z_{0} + tν_{Z}(z_{0}) : t > 0} is tangent to X at x_{0}. Then the set Z_{0} of those z ∈ Z such that the corresponding ray θ(z) is tangent to X at some point x(z) ∈ θ(z) is an (n − 2)-dimensional submanifold of Z locally near z_{0}. Consequently, the set X_{0} = {x(z) : z ∈ Z_{0}} is an (n − 2)-dimensional submanifold of X locally near x_{0}.

Proof. Let λ_{0} = (z_{0}, ν_{Z}(z_{0})) ∈ S_{Z}^{∗}(IR^{n}) and σ_{0} = (x_{0}, ν_{Z}(z_{0})) ∈ S_{Z}^{∗}(IR^{n}). Consider a defining function φ of X near x_{0} with ∇φ = ν_{X}, and let M = {σ ∈ U_{0} : p(σ) = H_{p}(σ) = 0}, where U_{0} is a small open neighbourhood of σ_{0} in T^{∗}(IR^{n}). Then M is a symplectic submanifold of T^{∗}(IR^{n}) of codimension 2. Consider the local map Λ : S_{Z}^{∗}(IR^{n}) ⟶ M whoich shifts each σ ∈ S_{Z}^{∗}(IR^{n}) along the free flow of H_{p} in T^{∗}(IR^{n}) until it hits M. That this map is well-defined, smooth and symplectic near λ_{0} follows in a standard way (see e.g. the proof of Lemma 5.2 above). As in Lemma 5.2, set M_{0} = {σ ∈ M : φ(σ) = 0}; then Λ^{−1}(M_{0}) is a codimension 1 submanifold of S_{Z}^{∗}(IR^{n}) (near λ_{0}). We claim that Λ^{−1}(M_{0}) is transversal in S_{Z}^{∗}(IR^{n}) to the (n − 1)-dimensional (Lagrangian) submanifold X̃. This is in fact trivial: consider a short strightline segment at y_{0} transversal to Y, for example r(s) = x_{0} + sν_{X}(x_{0}), |s| ≤ a, for some small a. Since Z is convex, the orthogonal projection to Z on a small open neighbourhood of x_{0} in IR^{n} is smooth and has maximal rank. Thus, the image z(s) of r(s) under this projection is a smooth curve on Z with z(0) = z_{0}. Moreover the curve σ(s) = (z(s), ν_{Z}(z(s)) in Z̃ is not tangent to Λ^{−1}(M_{0}) at z_{0}. Hence Z̃ is transversal to Λ^{−1}(M_{0}) at z_{0}. Since Z_{0} = pr_{1}(Z̃ ∩ Λ^{−1}(M_{0})), it follows that locally near z_{0}, Z_{0} is a smooth (n − 2)-dimensional submanifold of Z.

The statement about X_{0} follows from the fact that Λ(Z̃ ∩ Λ^{−1}(M_{0})) = Λ(Z̃) ∩ M_{0} is a smooth (n − 2)-dimensional submanifold of M_{0} and locally near σ_{0} the projection pr_{1} : M_{0} ⟶ IR^{n} has maximal rank.

Given σ ∈ S^{∗}(Ω_{K}) we denote γ_{K}^{+}(σ) = {pr_{1}(F_{t}^{(K)}(σ)) : t ≥ 0}.

We are going to use the construction in the proof of the above lemma to show that for obstacles K of the form (3) there are relatively few points σ ∈ S^{∗}(C) that generate trajectories with more than one tangency to ∂K. Here as before C is the boundary sphere of a large open ball U containing K in its interior.

Proposition 5.4. (a) There exists a countable family {R_{i}} of codimension 2 smooth submanifolds of S^{∗}(C) such that for any σ ∈ S^{∗}(C) \ (∪_{i}R_{i}) the trajectory γ_{K}(σ) has at most one tangency to ∂K.

(b) There exists a countable family {R_{i}} of codimension 2 smooth submanifolds of S_{∂K}^{∗}(Ω_{K}) such that for any σ ∈ S^{∗}(C) \ (∪_{i}R_{i}) the trajectory γ_{K}(σ) has at most one tangency to ∂K.

Proof. (a) Denote by Σ the set of those σ ∈ S^{∗}(C) such that γ_{K}(σ) has at least two tangencies to ∂K. Given integers 1 ≤ p < q, let Σ(p, q) be the set of those σ ∈ Σ such that γ_{K}(σ) has at least two different tangent points to ∂K, the first of which has the form F_{s_{1}}(σ) for some s_{1} ∈ [p\frac{d}{4}, (p+1)\frac{d}{4}] and the second one has the form F_{s_{2}}(σ) for some s_{2} ∈ [q \frac{d}{4}, (q +1)\frac{d}{4}]. Clearly for such σ, s_{1} = s_{1}(σ) and s_{2}(σ), we have that if ρ ∈ Σ(p, q) is close to σ, then F_{s_{1}(ρ)}(ρ) and F_{s_{1}(σ)}(σ) belong to the same convex component of K and the same applies to F_{s_{2}(ρ)}(ρ) and F_{s_{2}(σ)}(σ).

It is enough to show that for any p and q, the set Σ(p, q) can be covered by a countable union of codimension 2 submanioflds of S^{∗}(C). For this of course it is enough to prove the corresponding local statement: every σ ∈ Σ(p, q) has an open neighbourhood U in S^{∗}(C) such that Σ(p, q) ∩ U is contained in a countable union of codimension 2 submanifolds of U.

Fix integers q > p ≥ 1 and an arbitrary σ′ ∈ Σ(p, q). Set s_{1} = s_{1}(σ′), s_{2} = s_{2}(σ′), σ_{0} = F_{s_{1}}(σ′) and ρ_{0} = F_{s_{2}}(σ′). Then σ_{0}, ρ_{0} ∈ S^{∗}(∂K) and ρ_{0} = F_{t_{0}}(σ_{0}) for t_{0} = s − 2 − s_{1} > 0, so we can apply Lemma 5.2. The latter gives that there exist open neighbourhoods U_{0} and W_{0} of σ_{0} and ρ_{0} in T^{∗}(IR^{n}) such that if M_{0} ⊂ M ⊂ M′, N_{0} ⊂ N ⊂ N′ and the map Φ : M′ ⟶ N′ are defined as before the statement of Lemma 5.2, then M_{0} ∩ Φ^{−1}(N_{0}) is a submanifold of M_{0} of positive codimension. Consequently, the codimension of M_{0} ∩ Φ^{−1}(N_{0}) in M is at least 2. Let k be the number of (transversal) reflection points of {F_{t}(σ′) : 0 ≤ t < s_{1}}. Take a small open neighbourhood U′ of σ′ in T_{C}^{∗}(IR^{n}) and define Φ′ : U′ ⟶ M′ similarly to Φ: given σ ∈ U′, we consider the shift of σ′ along the flow F_{t} until after the kth reflection point, and after that the shift continuous along the free flow of H_{p} in IR^{n} until the trajectory hits N′. Clearly, if U′ is small enough, then Φ′ is well-defined and induces a symplectic map Φ′ : U = U′ ∩ S^{∗}(C) ⟶ M. Consequently, (Φ′)^{−1}(M_{0} ∩ Φ^{−1}(N_{0})) is a submanifold of U of codimension at least 2. Since Σ(p, q) ∩ U ⊂ M_{0} ∩ Φ^{−1}(N_{0}), the assertion is proved.

(b) The proof is almost the same as that of part (a). Define Σ(p, q) as before, however now p < q can be negative or 0 as well. If σ′ ∈ Σ(p, q) ∩ S^{∗}(∂K), one can simply apply Lemma 5.2. If σ′ ∈ Σ(p, q) ∩ S_{∂K}^{∗}(Ω_{K}), then one can apply the argument in the proof of part (b), replacing S^{∗}(C) by S_{∂K}^{∗}(Ω_{K}). Finally, if pr_{1}(σ′) ∉ ∂K, one take any transversal cros-section U′ of T^{∗}(Ω_{K}) at σ′ and proceed as in the proof of part (a). We omit the details.

As a consequence of the above considerations we get some information about the size of the set Trap(Ω_{K}) in the case when K has the form (3). We will denote by Trap^{+}(K) the set of those σ ∈ S^{∗}(Ω_{K}) such that the forward trajectory γ_{K}^{+}(σ) is bounded.

Proposition 5.5. Let n ≥ 3, let K have the form (3), and let U be an open ball containing K and C = ∂U. Then the subspace S^{∗}(C) \ Trap(Ω_{K}) of S^{∗}(C) is connected.

Proof. Clearly it is enough to prove that if U is a sufficiently large open ball, then S^{∗}(IR^{n} \U)\ Trap(Ω_{K}) is connected. Take an arbitrary σ_{0} = (x_{0}, ξ_{0}) ∈ S^{∗}(IR^{n} \ U) \ Trap(Ω_{K}). It is enough to find a continuous curve σ(t) in S^{∗}(IR^{n} \ U) \ Trap(Ω_{K}) such that σ(0) = σ_{0} and σ(1) generates a free trajectory in Ω_{K}, i.e. a trajectory without any reflections at ∂K.

We will assume that the trajectory γ_{K}(σ_{0}) has common points with S_{∂K}^{∗}(Ω_{K}); otherwise there is nothing to prove. Replacing ξ_{0} by −ξ_{0} if necessary, we will assume that γ_{K}^{+}(σ_{0}) has common points with S_{∂K}^{∗}(Ω_{K}). Then replacing x_{0} by x_{0} − tξ_{0} for some large t > 0, we may assume that there exists a strictly convex smooth hypersurface X in IR^{n} \U with a continuous unit normal field ν_{X} such that x_{0} ∈ X, ν_{X}(x_{0}) = ξ_{0}, and there exists x_{1} ∈ X such that the ray {x_{1}+tν_{X}(x_{1}) : t > 0} has no common points with K. (For X one can take the boundary sphere of a very large ball in IR^{n} \ U.) Take a very large closed ball U_{1} that contains the orthogonal projection of K onto X and the point x_{1} as well. Since the set Trap(Ω_{K}) ∩ S^{∗}(U_{1}) is compact and σ_{0} ∉ Trap(Ω_{K}), there exists and open connected neighbourhood V_{0} of σ_{0} in S^{∗}(U_{1}) with V_{0} ∩ Trap(Ω_{K}) = ∅.

It follows from Proposition 5.4 (a) that there exists a countable family {Q_{i}} of smooth codimension 2 submanifolds of S^{∗}(IR^{n} \ U) such that for any σ ∈ S^{∗}(IR^{n} \ U) \ (∪_{i}Q_{i}), the trajectory γ_{K}(σ) has at most one tangency to ∂K. The submanifolds Q_{i} are obtained from the submanifolds IR_{i} in Proposition 5.4 (a) by translation along the second (vector) component. So, locally Q_{i} are invariant under F.

Using Thom’s Transversality Theorem (cf. e.g. [H]), we can find an arbitrarily small in the C^{∞} Whitney topology deformation Y of X such that Ỹ is transversal to Q_{i} in S^{∗}(V) for each i. Since each Q_{i} is locally invariant under F_{t} (for small |t|), we may assume that x_{0} ∈ Y. Moreover, taking Y sufficiently close to X, we may assume that Y ∩ U_{1} is strictly convex and if η_{0} is the normal to Y at x_{0}, then (x_{0}, η_{0}) ∈ V_{0}, so (x_{0}, η_{0}) and (x_{0}, ξ_{0}) can be joined by a continuous curve in V_{0} which therefore has no common points with Trap(Ω_{K}). Finally, we can take Y so close to X that there exists y_{1} ∈ Y (close to x_{1}) such that the ray {y_{1} + tν_{Y} (y_{1}) : t > 0} has no common points with K.

The above argument shows that without loss of generality we may assume that X̃ is transversal to each of the submanifolds Q_{i}. Since dim(Q_{i}) = 2n − 3 and dim(X̃) = n − 1, it then follows that for each i, X̃ ∩Q_{i} is a submanifold of S^{∗}(V) with dim(X̃ ∩Q_{i}) = (2n−3+n−1)−(2n−1) = n−3. Using the Sum Theorem for the topological dimension dim (cf. e.g. [HW]), it now follows that for X′ = X̃ ∩ (∪_{i}Q_{i}) we have dim(X′) ≤ n − 3.

Next, denote by X_{0} the set of those σ ∈ X̃ ∩ Trap(Ω_{K}) such that the trajectory γ_{K}(σ) has no tangencies to ∂K. Given integers p, q such that q ∈ {1, …, s} and p ≥ 1, denote by X(p, q) the set of those σ ∈ X̃ ∩ Trap(Ω_{K}) such that γ_{K}(σ) has exactly one tangency to ∂K which is its pth reflection point and it belongs to ∂K_{q}. We will show that each of the subspaces X_{0} and X(p, q) of X̃ has topological dimension dim = 0.

Let F̃ = ∏_{r=1}^{∞} F be endowed with the product topology, where F = {1, 2, …, s} with the discrete topology. It is well known (and easy to see) that dim(F̃) = 0 and therefore every subspace of F̃ has topological dimension zero (cf. e.g. [HW]). Consider the map f : X_{0} ⟶ F̃, defined by f(σ) = (i_{1}, i_{2}, …), where the jth reflection point of γ_{K}(σ) belongs to ∂K_{i_{j}} for all j = 1, 2, . . .. Clearly, the map f is continuous and it follows from [St1] that f^{−1} : f(X_{0}) ⟶ X_{0} is also continuous (it is in fact Lipschitz with respect to an appropriate metric on F̃). Thus, X_{0} is homeomorphic to a subspace of F̃ and therefore dim(X_{0}) = 0.

Given p, q, we deal with X(p, q) in a similar way. Define f : X(p, q) ⟶ F̃ as above and notice that in the present case we have i_{q} = p for any σ ∈ X(p, q), where f(σ) = (i_{j}). This and the definition of X(p, q) imply that f is continuous on X(p, q) and using [St1] again, we get that X(p, q) is homeomorphic to a subspace of F̃ and so dim(X(p, q)) = 0.

Now the Sum Theorem for dim (cf. [HW]) shows that dim(X′ ∪ X_{0} ∪ ∪_{p,q}X(p, q)) ≤ n − 3. Since X̃ is an (n − 1)-dimensional Cantor manifold (cf. Sect. 6.5 in [HW]), it follows that X̃ \ Trap(Ω_{K}) is connected. Since σ_{0} and σ_{1} = (x_{1}, ν_{X}(x_{1})) both belong to X̃ \ Trap(Ω_{K}), there exists a continuous curve σ(t) in X̃ \ Trap(Ω_{K}) with σ(0) = σ_{0} and σ(1) = σ_{1}. This proves the assertion.

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Max-Planck-Institute für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

and

Department of Mathematics, University of Western Australia, Perth 6907 W. A. E-mail: stoyanov@maths.uwa.edu.au