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

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.


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