Translated points and Rabinowitz Floer homology

Abstract

We prove that if a contact manifold admits an exact filling then every local contactomorphism isotopic to the identity admits a translated point in the interior of its support, in the sense of Sandon [San11b]. In addition we prove that if the Rabinowitz Floer homology of the filling is non-zero then every contactomorphism isotopic to the identity admits a translated point, and if the Rabinowitz Floer homology of the filling is infinite dimensional then every contactmorphism isotopic to the identity has either infinitely many translated points, or a translated point on a closed leaf. Moreover if the contact manifold has dimension greater than or equal to 3, the latter option generically doesn't happen. Finally, we prove that a generic contactomorphism on ${\mathbb{R}}^{2n+1}$ has infinitely many geometrically distinct iterated translated points all of which lie in the interior of its support.

13 pages, v2: numerous corrections, results unchanged

References in corpus (3)

• Rabinowitz Floer homology: A survey
• On iterated translated points for contactomorphisms of R^{2n+1} and R^{2n} x S^1
• A variational approach to Givental's nonlinear Maslov index

Cited by in corpus (9)

• Floer homology of automorphisms of Liouville domains
• Orderability and the Weinstein Conjecture
• Isometry-invariant geodesics and the fundamental group
• Bounds for tentacular Hamiltonians
• Positive loops and $L^{\infty }$-contact systolic inequalities
• Rabinowitz Floer homology of negative line bundles and Floer Gysin sequence
• Bi-invariant metrics on contactomorphism groups
• Relative growth rate and contact Banach-Mazur distance
• Generating functions in symplectic and contact geometry