Volume entropy

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.
Definition.
Let ("M", "g") be a compact Riemannian manifold, with universal cover formula_1 Choose a point formula_2.
The volume entropy (or asymptotic volume growth) formula_3 is defined as the limit
where "B"("R") is the ball of radius "R" in formula_5 centered at formula_6 and "vol" is the Riemannian volume in the universal cover with the natural Riemannian metric.
A. Manning proved that the limit exists and does not depend on the choice of the base point. This asymptotic invariant describes the exponential growth rate of the volume of balls in the universal cover as a function of the radius.
Application in differential geometry of surfaces.
Katok's entropy inequality was recently exploited to obtain a tight asymptotic bound for the systolic ratio of surfaces of large genus, see systoles of surfaces.