First Steps in Twisted Rabinowitz-Floer Homology

TL;DR

This paper introduces a generalization of Rabinowitz-Floer homology to Liouville automorphisms, enabling new results on periodic Reeb orbits on symmetric star-shaped hypersurfaces and lens spaces.

Contribution

It extends Rabinowitz-Floer homology to a broader class of symplectic automorphisms, providing new tools for studying Reeb dynamics on quotient spaces.

Findings

Existence of noncontractible periodic Reeb orbits on quotients of symmetric star-shaped hypersurfaces.

Application of the generalized theory to lens spaces.

Development of a new Floer homology framework for Liouville automorphisms.

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 our work, 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, our theory applies to lens spaces.