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Growth of the number of geodesics between points and insecurity for Riemannian manifolds arXiv:math.DS/0701579
"... (Communicated by Anatole Katok) Abstract. A Riemannian manifold is said to be uniformly secure if there is a finite number s such that all geodesics connecting an arbitrary pair of points in the manifold can be blocked by s point obstacles. We prove that the number of geodesics with length ≤ T betwe ..."
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(Communicated by Anatole Katok) Abstract. A Riemannian manifold is said to be uniformly secure if there is a finite number s such that all geodesics connecting an arbitrary pair of points in the manifold can be blocked by s point obstacles. We prove that the number of geodesics with length ≤ T between every pair of points in a uniformly secure manifold grows polynomially as T → ∞. By results of Gromov and Mañé, the fundamental group of such a manifold is virtually nilpotent, and the topological entropy of its geodesic flow is zero. Furthermore, if a uniformly secure manifold has no conjugate points, then it is flat. This follows from the virtual nilpotency of its fundamental group either via the theorems of CrokeSchroeder and BuragoIvanov, or by more recent work of Lebedeva. We derive from this that a compact Riemannian manifold with no conjugate points whose geodesic flow has positive topological entropy is totally insecure: the geodesics between any pair of points cannot be blocked by a finite number of point obstacles.
INSECURITY FOR COMPACT SURFACES OF POSITIVE GENUS
, 908
"... 2. Rays, corays, and Busemann functions 4 3. Outline of the proof that nonflat twotori are insecure 6 4. Minimal geodesics in an admissible strip 9 ..."
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2. Rays, corays, and Busemann functions 4 3. Outline of the proof that nonflat twotori are insecure 6 4. Minimal geodesics in an admissible strip 9
A dense Gdelta set of Riemannian metrics without the finite blocking property
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Birkhoff billiards are insecure
"... We prove that every compact plane billiard, bounded by a smooth curve, is insecure: there exist pairs of points A,B such that no finite set of points can block all billiard trajectories from A to B. Two points A and B of a Riemannian manifold M are called secure if there exists a finite set of point ..."
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We prove that every compact plane billiard, bounded by a smooth curve, is insecure: there exist pairs of points A,B such that no finite set of points can block all billiard trajectories from A to B. Two points A and B of a Riemannian manifold M are called secure if there exists a finite set of points S ⊂ M − {A, B} such that every geodesic connecting A and B passes through a point of S. One says that the set S blocks A from B. A manifold is called secure (or has the finite blocking property) if any pair of its points is secure. For example, every pair of nonantipodal points of the Euclidean sphere is secure, but a pair of antipodal points is not secure, so the sphere is insecure. A flat torus of any dimension is secure. In the recent years, the notion of security has attracted a considerable attention, see [1, 2, 3, 4, 9, 10, 11, 12]. This notion extends naturally to Riemannian manifolds with boundary, in which case one considers billiard trajectories from A to B with the billiard reflection off the boundary. In this note we consider a compact plane billiard domain M bounded by a smooth curve and prove that M is insecure. More specifically, one has the following local insecurity result. Consider a sufficiently short outward convex arc γ ⊂ ∂M with endpoints A and B (such an arc always exists).
Blocking: New examples and properties of products
"... Abstract. We say that a pair of points x and y is secure if there exist a finite set of blocking points such that any geodesic between x and y passes through one of the blocking points. The main point of this paper is to exhibit new examples of blocking phenomena both in the manifold and the billiar ..."
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Abstract. We say that a pair of points x and y is secure if there exist a finite set of blocking points such that any geodesic between x and y passes through one of the blocking points. The main point of this paper is to exhibit new examples of blocking phenomena both in the manifold and the billiard table setting. As an approach to this, we study if the product of secure configurations (or manifolds) is also secure. We introduce the concept of midpoint security that imposes that the geodesic reaches a blocking point exactly at its midpoint. We prove that products of midpoint secure configurations are midpoint secure. On the other hand, we give an example of a compact C 1 surface that contains secure configurations that are not midpoint secure. This surface provides the first example of an insecure product of secure configurations, as well as billiard table examples. 1.
Topological entropy and blocking cost for geodesics in Riemannian manifolds
, 2007
"... Abstract. For a pair of points x, y in a compact, riemannian manifold M let nt(x, y) (resp. st(x, y)) be the number of geodesic segments with length ≤ t joining these points (resp. the minimal number of point obstacles needed to block them). We study relationships between the growth rates of nt(x, y ..."
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Abstract. For a pair of points x, y in a compact, riemannian manifold M let nt(x, y) (resp. st(x, y)) be the number of geodesic segments with length ≤ t joining these points (resp. the minimal number of point obstacles needed to block them). We study relationships between the growth rates of nt(x, y) and st(x, y) as t → ∞. We derive lower bounds on st(x, y) in terms of the topological entropy h(M) and its fundamental group. This strengthens the results of BurnsGutkin [2] and LafontSchmidt [13]. For instance, by [2, 13], h(M)> 0 implies that s is unbounded; we show that s grows exponentially, with the rate at least h(M)/2.
Real analytic metrics on S2 with total absence of finite blocking, preprint arXiv:1109.1336
, 2011
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A CHARACTERIZATION OF ROUND SPHERES IN TERMS OF BLOCKING LIGHT
, 704
"... Abstract. A closed Riemannian manifold M is said to have cross (compact rank one symmetric space) blocking if whenever p = q are less than the diameter apart, all light rays from p can be shaded away from q with at most two point shades. Similarly, a closed Riemannian manifold is said to have spher ..."
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Abstract. A closed Riemannian manifold M is said to have cross (compact rank one symmetric space) blocking if whenever p = q are less than the diameter apart, all light rays from p can be shaded away from q with at most two point shades. Similarly, a closed Riemannian manifold is said to have sphere blocking if for each p ∈ M all the light rays from p are shaded away from p by a single point shade. We prove that Riemannian manifolds with cross and sphere blocking are isometric to round spheres. In this note we characterize constant curvature spheres in terms of light blocking properties. Definition (Light). Let X, Y be two nonempty subsets of a Riemannian manifold M, and let GM(X, Y) denote the set of nonconstant unit speed parametrized geodesics γ: [0, Lγ] → M with initial point γ(0) ∈ X and terminal point γ(Lγ) ∈ Y. The light from X to Y is the set LM(X, Y) = {γ ∈ GM(X, Y)  interior(γ) ∩ (X ∪ Y) = ∅}. A subset Z ⊂ M blocks the light from X to Y if the interior of every γ ∈ LM(X, Y) meets Z. Intuitively, we are postulating that X emits light traveling along geodesics, that Y consists of receptors, and that X and Y are opaque while the remaining medium M \ {X ∪ Y} is transparent. From this point of view, LM(X, Y) is the set of light rays from X to Y and a set Z blocks the light from X to Y if it completely shades X away from Y. This simple model ignores diffraction, the dual nature of light, and all aspects of quantum mechanics. A well known result of Serre [Se51] asserts that for compact M and points x, y ∈ M, the set GM(x, y) of geodesic segments joining x and y is always infinite. In contrast, LM(x, y) is sometimes infinite and sometimes not. For instance, if x and y are different points on the standard round sphere Sn with distance less than π, then LSn(x, y) consists of exactly two elements. In particular, we see that, under the 1 2 BENJAMIN SCHMIDT & JUAN SOUTO same assumptions, it suffices to declare two additional points in S n to be opaque in order to block all the light rays from x to y.
CHORDS, LIGHT, AND ANOTHER SYNTHETIC CHARACTERIZATION OF THE ROUND SPHERE
, 704
"... Abstract. A chord for a closed geodesic γ in a complete Riemannian manifold M is a nontrivial geodesic segment beginning and ending on γ that is not completely contained in γ. We prove the existence of at least one geodesic chord for every closed geodesic in a closed Riemannian manifold. As an appli ..."
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Abstract. A chord for a closed geodesic γ in a complete Riemannian manifold M is a nontrivial geodesic segment beginning and ending on γ that is not completely contained in γ. We prove the existence of at least one geodesic chord for every closed geodesic in a closed Riemannian manifold. As an application, we give a synthetic characterization of round spheres in terms of blocking light. The study of closed geodesics in Riemannian manifolds has a long and rich history. In compact manifolds with nontrivial fundamental group, closed geodesics are at least as plentiful as free homotopy classes; namely, homotopically essential curves can be pulled tight to closed geodesics. For compact, simply connected manifolds, more sophisticated techniques are needed to prove the existence of closed geodesics. In the 1930’s Lyusternik and Shnirelman (see [Ba78]) proved that every closed simply connected manifold contains at least 3 geometrically
SECURE TWODIMENSIONAL TORI ARE FLAT
, 806
"... Abstract. A riemannian manifold M is secure if the geodesics between any pair of points in M can be blocked by a finite number of point obstacles. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds. The conjecture claims, in pa ..."
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Abstract. A riemannian manifold M is secure if the geodesics between any pair of points in M can be blocked by a finite number of point obstacles. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds. The conjecture claims, in particular, that a riemannian torus of any dimension is secure if and only if it is flat. We prove this for twodimensional tori. 1.