Question. Is there any example of a path connected symplectic manifold $(M,\omega)$ that has infinite volume, but which cannot be packed by an infinite number of symplectic balls of a fixed radius $r$, for any $r > 0$?
Note that on such a manifold we would be able to prove the "Poincaré recurrence" of every symplectomorphism:
Given a symplectomorphism $T: M \rightarrow M$ and an open set $U \subset M$ there exists an integer number $k > 0$ such that $T^k(U)$ intersects $U$.
Added and edited: It seems to me that the examples suggested in the comments assume the validity of the following (to my knowledge unproved) statement:
Let $(M,\omega)$ be the symplectic manifold obtained by taking $B(R)$, the symplectic ball of radius $R$ and dimension 2n, and attaching a long thin cylinder with very large volume and very small capacity. Let $r < R$ and consider the ball $B(r)$. There exists constant $c > 0$ such that any symplectic embedding of $B(r)$ into $M$ intersects the ball $B(R) \subset M$ in a set whose volume is at least $c$.
Note that this is not quite the intuition of non-squeezing theorem which would just say that you cannot embedd $B(r)$ into $M$ if $r > R$.
Added remark (10/10/2013). I just had a conversation around this problem with Leonid Polterovich. He tells me that in the early nineties Hofer had asked him whether symplectic geometry could perhaps be used to sharpen the Poincaré recurrence theorem. This question was part of the motivation for his work with McDuff on packing obstructions.
