The volume entropy is an asymptotic invariant of a compact Riemannian manifold that measures the exponential growth rate of the volume of metric balls in its universal cover. This concept is closely related with other notions of entropy found in dynamical systems and plays an important role in differential geometry and geometric group theory. If the manifold is nonpositively curved then its volume entropy coincides with the topological entropy of the geodesic flow. It is of considerable interest in differential geometry to find the Riemannian metric on a given smooth manifold which minimizes the volume entropy, with locally symmetric spaces forming a basic class of examples.
Read more about Volume Entropy: Definition, Properties, Application in Differential Geometry of Surfaces
Famous quotes containing the words volume and/or entropy:
“So it is with books, for the most part: they work no redemption on us. The bookseller might certainly know that his customers are in no respect better for the purchase and consumption of his wares. The volume is dear at a dollar, and after to reading to weariness the lettered backs, we leave the shop with a sigh, and learn, as I did without surprise of a surly bank director, that in bank parlors they estimate all stocks of this kind as rubbish.”
—Ralph Waldo Emerson (1803–1882)
“Just as the constant increase of entropy is the basic law of the universe, so it is the basic law of life to be ever more highly structured and to struggle against entropy.”
—Václav Havel (b. 1936)