Topological entropy for geodesic flows under Ricci curvature and conjugate radius bounded below (Q2752850)

Topological entropy is a measure of the total exponential complexity of the orbit structure of a dynamical system. In the case of compact Riemannian manifolds, the topological entropy of the geodesic flow is closely related to the volume growth rate of spheres of increasing radius. This paper presents two results: First, if the Ricci curvature of a Riemannian manifold is bounded from below and the conjugate radius is greater than some positive number, then the topological entropy has an upper bound that depends only on the dimension of the manifold and on the bounds of the Ricci curvature and conjugate radius. Second, when the bounds on the Ricci curvature and conjugate radius go to zero and \(+ \infty\), respectively, (satisfying a proportionality condition) the topological entropy must go to zero. The proof relies on Brocks's estimate of the Laplacian of the distance function.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00035].