Contact non-squeezing at large scale via generating functions

Abstract

Using SFT techniques, Eliashberg, Kim and Polterovich (2006) proved that if $\ddot{I}€R_2^2\le K\le \ddot{I}€R_1^2$ 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 $1\le \ddot{I}€R_2^2\le \ddot{I}€R_1^2$ 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.

42 pages. Revised version, to appear in the Journal of Fixed Point Theory and Applications