Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Growth of the number of geodesics between points and insecurity for Riemannian manifolds

Pages: 403 - 413, Volume 21, Issue 2, June 2008      doi:10.3934/dcds.2008.21.403

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Keith Burns - Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, United States (email)
Eugene Gutkin - IMPA, Estrada Dona Castorina 110, Rio de Janeiro 22460-320, Brazil (email)

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 $\leq T$ between every pair of points in a uniformly secure manifold grows polynomially as $T \to \infty$. 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 Croke-Schroeder and Burago-Ivanov, 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.

Keywords:  security, entropy, connecting geodesics.
Mathematics Subject Classification:  Primary: 53C22; Secondary: 37D40.

Received: May 2007;      Revised: October 2007;      Available Online: March 2008.