The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds. (English) Zbl 0946.53045

Knieper, Gerhard

Ann. Math. (2) 148, No. 1, 291-314 (1998).

Author’s abstract: “We prove a conjecture of A. Katok [K. Burns and A. Katok, Ergodic Theory Dyn. Syst. 5, 307-317 (1985; Zbl 0572.58019)] stating that the geodesic flow on a compact rank 1 manifold admits a uniquely determined invariant measure of maximal entropy. This generalizes previous work of R. Bowen [Math. Systems Theory 7, 300-303 (1974; Zbl 0303.58014)] and G. A. Margulis [Funct. Anal. Appl. 3, 335-336 (1969; Zbl 0207.20305)]. As an application we show that the exponential growth rate of the set of singular closed geodesics is strictly smaller than the topological entropy.

MSC:

53D25	Geodesic flows in symplectic geometry and contact geometry
58D20	Measures (Gaussian, cylindrical, etc.) on manifolds of maps

Keywords:

singular closed geodesic; geodesic flow; invariant measure; exponential growth rate; topological entropy

Citations:

Zbl 0572.58019; Zbl 0303.58014; Zbl 0207.20305