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

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 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 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 R^{2n}, and is obtained with generating functions techniques in the setting of arXiv:0901.3112.

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

References in corpus (4)

• Proof of the Arnold chord conjecture in three dimensions II
• Cup-length estimates for leaf-wise intersections
• Spectral Invariants in Rabinowitz Floer homology and Global Hamiltonian perturbations
• On the Rabinowitz Floer homology of twisted cotangent bundles

Cited by in corpus (9)

• Floer homology of automorphisms of Liouville domains
• Translated points and Rabinowitz Floer homology
• Orderability and the Weinstein Conjecture
• Givental's non-linear Maslov index on lens spaces
• Isometry-invariant geodesics and the fundamental group
• Coisotropic Ekeland-Hofer capacities
• Bi-invariant metrics on contactomorphism groups
• Rabinowitz Floer homology of negative line bundles and Floer Gysin sequence
• Generating functions in symplectic and contact geometry