Topological entropy and blocking cost for geodesics in riemannian manifolds

TL;DR

This paper explores the relationship between the growth of geodesic counts and obstacle blocking costs in Riemannian manifolds, establishing exponential growth bounds linked to topological entropy.

Contribution

It provides new lower bounds on blocking costs based on topological entropy, strengthening previous results and revealing exponential growth rates.

Findings

Blocking cost grows at least exponentially with rate h(M)/2.

Topological entropy h(M) influences geodesic complexity.

Results extend prior work by Burns-Gutkin and Lafont-Schmidt.

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