TY - JOUR
T1 - Counting geodesics on a Riemannian manifold and topological entropy of geodesic flows
AU - Burns, Keith
AU - Paternain, Gabriel P.
PY - 1997/10
Y1 - 1997/10
N2 - Let M be a compact C∞ Riemannian manifold. Given p and q in M and T > 0, define nT(p, q) as the number of geodesic segments joining p and q with length ≤ T. Mañe showed in [7] that lim T→∞ 1/T log ∫ M×M nT(p,q) dpdq = htop, where htop denotes the topological entropy of the geodesic flow of M. In this paper we exhibit an open set of metrics on the two-sphere for which lim sup T→∞ 1/T log nT (p, q) < htop, for a positive measure set of (p, q) ∈ M × M. This answers in the negative questions raised by Mañé in [7].
AB - Let M be a compact C∞ Riemannian manifold. Given p and q in M and T > 0, define nT(p, q) as the number of geodesic segments joining p and q with length ≤ T. Mañe showed in [7] that lim T→∞ 1/T log ∫ M×M nT(p,q) dpdq = htop, where htop denotes the topological entropy of the geodesic flow of M. In this paper we exhibit an open set of metrics on the two-sphere for which lim sup T→∞ 1/T log nT (p, q) < htop, for a positive measure set of (p, q) ∈ M × M. This answers in the negative questions raised by Mañé in [7].
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U2 - 10.1017/S0143385797086331
DO - 10.1017/S0143385797086331
M3 - Article
AN - SCOPUS:0039054224
VL - 17
SP - 1043
EP - 1059
JO - Ergodic Theory and Dynamical Systems
JF - Ergodic Theory and Dynamical Systems
SN - 0143-3857
IS - 5
ER -