In this paper we derive new asymptotic properties of all Hadamard manifolds admitting compact quotients. We study the growth function of the volume of geodesic spheres, generalizing the work of Margulis in the case of negative curvature. We show that the growth is of purely exponential type if and only if the Hadamard manifold is of rank 1. In general, there is a polynomial deviation from purely exponential behavior, depending in an unexpected way on the rank of the manifold. Furthermore, we obtain new results on the growth rate of closed geodesics on compact rank 1 spaces.
Summary. In this paper we investigate the regularity of the topological entropy hto p for C k perturbations of Anosov flows. We show that the topological entropy varies (almost) as smoothly as the perturbation. The results in this paper, along with several related results, have been announced in [KKPW].The authors wish to thank Bernard Shiffman for graciously supplying us with the proof of the key technical theorem used in step 3 of Theorem 1. Strategy of proof. We give two proofs of Theorem I. The first proof uses zeta functions and complex analysis and only works for C ~' perturbations. This proof yields valuable insight into how the periods of closed trajectories change when the *
Abstract. The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank 1. This conjecture has been proved by Z. Szabó [Sz] for harmonic manifolds with compact universal cover. E. Damek and F. Ricci [DR] provided examples showing that in the noncompact case the conjecture is wrong. However, such manifolds do not admit a compact quotient.In this paper we study, using a notion of rank, the asymptotic geometry and the geodesic flow on simply connected nonflat and noncompact harmonic manifolds denoted by X.In the first part of the paper we show that the following assertions are equivalent. The volume growth is purely exponential, the rank of X is one, the geodesic flow is Anosov with respect to the Sasaki metric, X is Gromov hyperbolic.In the second part of the paper we show that the geodesic flow is Anosov if X is a nonflat harmonic manifold with no focal points. In the course of the proof we obtain that certain partially hyperbolic flows on arbitrary Riemannian manifolds without focal points are Anosov, which is of interest beyond harmonic manifolds.Combining the results of this paper with the rigidity theorem's of [BCG] , [BFL] and [FL], we confirm the Lichnerowicz conjecture for all compact harmonic manifolds without focal points or with Gromov hyperbolic fundamental groups.
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