Translated points and Rabinowitz Floer homology

TL;DR

This paper establishes conditions under which contactomorphisms on certain contact manifolds have translated points, linking these points to properties of Rabinowitz Floer homology and demonstrating generic multiplicity results.

Contribution

It proves the existence of translated points for contactomorphisms under conditions related to exact fillings and Rabinowitz Floer homology, and shows generic multiplicity of such points in Euclidean space.

Findings

Existence of translated points for contactomorphisms with exact fillings.

Non-zero Rabinowitz Floer homology implies existence of translated points.

Generic contactomorphisms in $\,\mathbb{R}^{2n+1}$ have infinitely many distinct iterated translated points.

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.