Lectures on Twisted Rabinowitz-Floer Homology

TL;DR

This paper extends Rabinowitz-Floer homology to Liouville automorphisms, providing new tools for detecting periodic Reeb orbits and establishing existence results in symplectic and contact geometry.

Contribution

It generalizes Rabinowitz-Floer homology to include Liouville automorphisms and applies this to prove existence of periodic orbits on quotients of star-shaped hypersurfaces.

Findings

Existence of noncontractible periodic Reeb orbits on lens spaces.

A forcing theorem for contractible twisted closed characteristics.

Application to displaceable twisted stable hypersurfaces.

Abstract

Rabinowitz-Floer homology is the Morse-Bott homology in the sense of Floer associated with the Rabinowitz action functional introduced by Kai Cieliebak and Urs Frauenfelder in 2009. In this manuscript, we consider a generalisation of this theory to a Rabinowitz-Floer homology of a Liouville automorphism. As an application, we show the existence of noncontractible periodic Reeb orbits on quotients of symmetric star-shaped hypersurfaces. In particular, this theory applies to lens spaces. Moreover, we prove a forcing theorem, which guarantees the existence of a contractible twisted closed characteristic on a displaceable twisted stable hypersurface in a symplectically aspherical geometrically bounded symplectic manifold if there exists a contractible twisted closed characteristic belonging to a Morse-Bott component, with energy difference smaller or equal to the displacement energy of the displaceable hypersurface.