Definition
Let (M, g) be a compact Riemannian manifold, with universal cover Choose a point .
The volume entropy (or asymptotic volume growth) is defined as the limit
where B(R) is the ball of radius R in centered at 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.
Read more about this topic: Volume Entropy
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