# Poincaré recurrence and symplectic packings

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 "PoincarĂ© 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 PoincarĂ© recurrence theorem. This question was part of the motivation for his work with McDuff on packing obstructions.

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My intuition is that Gromov non-squeezing might make the following example work: Start with the stupid example of a sequence of disjoint balls with radius converging to 0, but with infinite volume. Then connect these by very thin tubes (thin in the sense that Gromov's non-squeezing theorem doesn't allow large symplectic balls to pass through them.) –  Brett Parker Oct 4 '13 at 5:17
Can I not take the infinite "cylinder" whose metric has decreasing ends as $\sim\frac{1}{ln(r)}$? (and then apply Gromov nonsqueezing) –  Chris Gerig Oct 4 '13 at 6:04
@BrettParker and Chris Gerig, I think you are implicitly assuming that because one cannot symplectically displace a ball through a thin tube, one cannot displace a large part of its volume through it. –  alvarezpaiva Oct 5 '13 at 21:02
I agree that my suggestion relies on this statement about not being able to pass a large volume of a ball through a thin tube. To me, this statement does not seem to follow from the usual argument for Gromov non-squeezing, and after thinking for a little while, I'm inclined to think that it is false. –  Brett Parker Oct 6 '13 at 1:10