The volume entropy h(g) of a closed Riemannian n-manifold (M, g) is defined as h(g) = lim_(R → ∞) 1/R log(Vol(B(x,R)) where B{x, R) is the ball of radius R around a fixed point x in the universal cover X. (For noiicompact M, see Section 6.2.) The number h{g) is independent of the choice of a;, and equals the topological entropy of the geodesic now on (M.g) when the curvature K(g) satisfies K(g) ≤ 0 (see [Ma]). Note that while the volume Vol(M,g) is not invariant under scaling the metric g, the normalized entropy ent(g) = h(g)~nVol(M,g) is scale invariant.
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机译：闭合黎曼流形（M，g）的体积熵h（g）定义为h（g）= lim_（R→∞）1 / R log（Vol（B（x，R））其中B { x，R）是在通用保护套X中围绕固定点x的半径R的球。（关于小巧的M，请参见第6.2节。）数h {g）与a的选择无关;并且等于拓扑当曲率K（g）满足K（g）≤0时，测地线的熵（Mg）满足（请参见[Ma]）。注意，尽管在缩放度量g时体积Vol（M，g）不变，但归一化熵ent（g）= h（g）〜nVol（M，g）是尺度不变的。
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