Rigidity and flexibility in p-adic symplectic geometry

Let n 2 be an integer and let p be a prime number. We prove that the analog of Gromov's non-squeezing theorem does not hold for p-adic embeddings: for any p-adic absolute value R, the entire p-adic space (Qₚ) ^2n is symplectomorphic to the p-adic cylinder Zₚ^2n (R) of radius R, showing a degree of flexibility which stands in contrast with the real case. However, some rigidity remains: we prove that the p-adic affine analog of Gromov's result still holds. We will also show that in the non-linear situation, if the p-adic embeddings are equivariant with respect to a torus action, then non-squeezing holds, which generalizes a recent result by Figalli, Palmer and the second author. This allows us to introduce equivariant p-adic analytic symplectic capacities, of which the p-adic equivariant Gromov width is an example.