# volume entropy for manifolds with boundary?

Recently I learn the volume entropy: http://en.wikipedia.org/wiki/Volume_entropy I wonder if one can define volume entropy for a manifold with boundary. Does anyone come up with a reasonable definition of it and obtain some results in this direction? Or it doesn't make sense at all to talk about it? If yes, why is that the case?

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Imagine that you can complete your riemannian manifold M with boundary into a connected riemannian manifold N without boundary, by gluing certain pieces, denoted by P.

Now, consider the universal cover of $N=M\cup P$. Call "universal cover of M" the universal cover of N minus all the lifts of P. (It is a priori not simply connected)

Now, you can define the volume entropy of M as usual, as the exponential growth rate of volume of balls, in the universal cover of M. (Even if the balls of this space are not simply connected, they are balls and have a volume)

I would guess that you still can prove that the volume entropy is an upper bound for the topological entropy of the geodesic flow, following Manning ideas.

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