1996/08/01 by Gérard Besson, Gilles Courtois, Sylvestre Gallot · 141 citations
Mathematics · #Mathematical Dynamics and Fractals #Geometric and Algebraic Topology #Geometric Analysis and Curvature Flows #Mathematics #Pure mathematics #Topological entropy #Amenable group #Entropy (arrow of time) #Invariant (physics) #Joint quantum entropy #Combinatorics #Mathematical physics #Principle of maximum entropy #Physics #Quantum mechanics
paper · doi:10.1017/s0143385700009019
published in Ergodic Theory and Dynamical Systems 16(4), 623-649 (Cambridge University Press)
openalex publication_date 1996/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22
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 ψ t , a flow on it, we define .