Orderability, contact non-squeezing, and Rabinowitz Floer homology

We study Liouville fillable contact manifolds $(\Sigma, \xi)$ with non-zero and spectrally finite Rabinowitz Floer homology and assign spectral numbers to paths of contactomorphisms. As a consequence we prove that $\widetilde{\mathrm{Cont}_0} (\Sigma, \xi)$ is orderable in the sense of Eliashberg and Polterovich. This provides a new class of orderable contact manifolds. If the contact manifold is in addition periodic or a prequantization space $M \times S^1$ for $M$ a Liouville manifold, then we construct a contact capacity in the sense of Sandon [44]. This can be used to prove a general non-squeezing result, which amongst other examples in particular recovers the beautiful non-squeezing results from Eliashberg, Kim, and Polterovich [24].

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Received 22 March 2016

Accepted 23 February 2018

Published 18 March 2019