Topological entropy and blocking cost for geodesics in riemannian manifolds

Details Topological entropy and blocking cost for geodesics in riemannian manifolds Topological entropy and blocking cost for geodesics in riemannian manifolds

2007-11-11

arxiv.org/abs/0711.1662v1

Geometriae Dedicata 138 (2009), 13 -- 23

Mathematics

Dynamical Systems

13 pages

Scientific paper

10.1007/s10711-008-9296-3

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 $\leq t$ 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 $h(M)/2$.

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