Topological entropy and blocking cost for geodesics in riemannian manifolds

Abstract

For a pair of points $x,y$ in a compact, riemannian manifold $M$ let $n_t(x,y)$ (resp. $s_t(x,y)$) be the number of geodesic segments with length $\le t$ joining these points (resp. the minimal number of point obstacles needed to block them). We study relationships between the growth rates of $n_t(x,y)$ and $s_t(x,y)$ as $t\to \infty$. We derive lower bounds on $s_t(x,y)$ in terms of the topological entropy $h(M)$ and its fundamental group. This strengthens the results of Burns-Gutkin \cite{BG06} and Lafont-Schmidt \cite{LS}. For instance, by \cite{BG06,LS}, $h(M)>0$ implies that $s$ is unbounded; we show that $s$ grows exponentially, with the rate at least $h(M)/2$.

13 pages

References in corpus (4)

• Blocking light in compact Riemannian manifolds
• On Blocking Numbers of Surfaces
• Secure two-dimensional tori are flat
• Blocking: New examples and properties of products

Cited by in corpus (3)

• Insecurity for compact surfaces of positive genus
• Billiard Dynamics: An Updated Survey with the Emphasis on Open Problems
• Connection blocking in homogeneous spaces and nilmanifolds