On the rigidity of translated points

TL;DR

This paper demonstrates the existence of contact isotopies on the standard contact sphere with no translated points near the identity, confirming the optimality of Shelukhin's theorem regarding translated points in certain contact manifolds.

Contribution

It establishes the sharpness of Shelukhin's theorem by constructing contact isotopies lacking translated points close to the identity in the Shelukhin-Hofer distance.

Findings

Existence of contact isotopies without close translated points

Sharpness of Shelukhin's theorem confirmed

Translated points can be absent near the identity in specific cases

Abstract

We show that there exist contact isotopies of the standard contact sphere whose time-1 maps do not have any translated points which are optimally close to the identity in the Shelukhin-Hofer distance. This proves the sharpness of a theorem of Shelukhin on the existence of translated points for contact isotopies of Liouville fillable contact manifolds with small enough Shelukhin-Hofer norm.