Contact non-squeezing at large scale via generating functions

Abstract

Using SFT techniques, Eliashberg, Kim and Polterovich (2006) proved that if $\pi R_2^2 \leq K \leq \pi R_1^2$ for some integer $K$ then there is no contact squeezing in $\mathbb{R}^{2n} \times S^1$ of the prequantization of the ball of radius $R_1$ into the prequantization of the ball of radius $R_2$. This result was extended to the case of balls of radius $R_1$ and $R_2$ with $1 \leq \pi R_2^2 \leq \pi R_1^2$ by Chiu (2017) and the first author (2016), using respectively microlocal sheaves and SFT. In the present article we recover this general contact non-squeezing theorem using generating functions, a classical method based on finite dimensional Morse theory. More precisely, we develop an equivariant version, with respect to a certain action of a finite cyclic group, of the generating function homology for domains of $\mathbb{R}^{2n} \times S^1$ defined by the second author (2011). A key role in the construction is played by translated chains of contactomorphisms, a generalization of translated points.