Ergodic Theory and Dynamical Systems

Survey Article

Minimal entropy and Mostow's rigidity theorems

Gérard Bessona1, Gilles Courtoisa2 and Sylvestre Gallota3

a1 Institut Fourier-C.N.R.S., U.R.A. 188, B.P. 74, 38402 Saint Martin d'Héres Cedex, France

a2 École Polytechnique-C.N.R.S., U.R.A. 169, Centre de Mathématiques, 91128 Palaiseau Cedex, France

a3 École Normale Supérieure de Lyon, U.M.R. 128, 46, Allée d'Italie, 69364 Lyon Cedex 07, France

Let (Y, g) be a compact connected n-dimensional Riemannian manifold and let () be its universal cover endowed with the pulled-back metric. If y , we define


where B(y, R) denotes the ball of radius R around y in . It is a well known fact that this limit exists and does not depend on y ([Man]). The invariant h(g) is called the volume entropy of the metric g but, for the sake of simplicity, we shall use the term entropy. The idea of recognizing special metrics in terms of this invariant looks at first glance very optimistic. First the entropy, which behaves like the inverse of a distance, is sensitive to changes of scale which makes it a bad invariant: however, this is a minor drawback that can be circumvented by looking at the behaviour of the entropy functional on the space of metrics with fixed volume (equal to one for example). Nevertheless, it seems very unlikely that two numbers, the entropy and the volume, might characterize any metric. The very first person to consider such a possibility was Katok ([Kat1]). In this article the entropy is thought of as a dynamical invariant which actually is suggested by its name. More precisely, let us define this dynamical invariant, which is called the topological entropy: let (M, d) be a compact metric space and ψt, a flow on it, we define