On iterated translated points for contactomorphisms of R^{2n+1} and R^{2n} x S^1

TL;DR

This paper proves that positive contactomorphisms on certain contact manifolds have infinitely many geometrically distinct iterated translated points, extending ideas from symplectic fixed point theory using generating functions.

Contribution

It establishes the existence of infinitely many iterated translated points for positive contactomorphisms, providing a contact analogue to Viterbo's fixed point theorem.

Findings

Existence of infinitely many iterated translated points for positive contactomorphisms.

Extension of symplectic fixed point results to contact geometry.

Application of generating functions techniques in contact topology.

Abstract

A point q in a contact manifold is called a translated point for a contactomorphism \phi, with respect to some fixed contact form, if \phi (q) and q belong to the same Reeb orbit and the contact form is preserved at q. The problem of existence of translated points is related to the chord conjecture and to the problem of leafwise coisotropic intersections. In the case of a compactly supported contactomorphism of R^{2n+1} or R^{2n} x S^1 contact isotopic to the identity, existence of translated points follows immediately from Chekanov's theorem on critical points of quasi-functions and Bhupal's graph construction. In this article we prove that if \phi is positive then there are infinitely many non-trivial geometrically distinct iterated translated points, i.e. translated points of some iteration \phi^k. This result can be seen as a (partial) contact analogue of the result of Viterbo on existence of infinitely many iterated fixed points for compactly supported Hamiltonian symplectomorphisms of R^{2n}, and is obtained with generating functions techniques in the setting of arXiv:0901.3112.