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?
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.