On iterated translated points for contactomorphisms of R²ⁿ⁺¹ and R²ⁿ x S¹

2011/02/28 by Sheila Sandon · 1 citation
Mathematics · #math.SG #math.GT #msc:53D35

paper · pdf · doi:10.1142/s0129167x12500425

10 pages, revised version. I removed the discussion on linear growth of iterated translated points, because it contained a mistake. To appear in the International Journal of Mathematics

arxiv created 2011/10/21 · arxiv updated 2012/06/19

Abstract

A point q in a contact manifold is called a translated point for a contactomorphism ϕ, with respect to some fixed contact form, if ϕ(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 R²ⁿ⁺¹ or R²ⁿ x S¹ 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 ϕis positive then there are infinitely many non-trivial geometrically distinct iterated translated points, i.e. translated points of some iteration ϕ^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 R²ⁿ, and is obtained with generating functions techniques in the setting of arXiv:0901.3112.

Citations

• Contact Homology, Capacity and Non-Squeezing in R²n x S¹ via Generating Functions

Cited by

• On flat translated chains of contactomorphisms of ℝ²ⁿ⁺¹ and ℝ²ⁿ × S¹