Closed geodesics and the uniqueness of the maximal measure for rank 1 geodesic flows (Q2778788)

Let \((M,g)\) be a compact \(n\)-dimensional Riemannian manifold of nonpositive curvature and \(X\) be its universal cover. The manifold \(M\) can be represented as \(X/\Gamma\), where \(\Gamma\) denotes the group of deck transformations which acts on \(X\). Further, \(B(p,r)\) and \(S(p,r)\) are the geodesic ball and sphere of radius \(r\) about \(p\in X\). The author gives an estimate for the volume growth of geodesic spheres in \(X\) in the following form: Let \(M= X/\Gamma\) be a compact rank one manifold and \(p\in X\). Then there exists a constant \(a> 1\) such that NEWLINE\[NEWLINE{1\over a}\leq {\text{vol }S(p, r)\over e^{hr}}\leq aNEWLINE\]NEWLINE for all \(r> 0\), where \(h\) denotes the entropy. Further, let \(P_{\text{reg}}(t)\) denote the set of regular primitive closed geodesics of period \(\leq t\) in \(M\). The author shows that there are constants \(A,B> 0\) such that \(B(e^{ht}/t)\leq \text{card}(P_{\text{reg}}(t))\leq A(e^{ht}/t)\). Finally, the author discusses some problems concerning the measure of maximal entropy. The paper is well written and the text is provided with suitable figures.NEWLINENEWLINEFor the entire collection see [Zbl 0973.00044].