Delayed Rabinowitz Floer Homology

TL;DR

This paper introduces a delayed version of Rabinowitz Floer Homology that maintains invariance under individual particle time translations and demonstrates that it is essentially equivalent to the classical version, with preserved critical points and actions.

Contribution

It defines a new delayed Rabinowitz action functional, shows its equivalence to the classical one, and establishes compactness of gradient flow lines under certain conditions.

Findings

Delayed Rabinowitz action functional is invariant under individual time translations.

Critical points of the delayed functional correspond to those of the classical functional.

Compactness of gradient flow lines is established under restricted contact type assumptions.

Abstract

In this article we study Rabinowitz Floer Homology for several interaction particles. In general Rabinowitz action functional is invariant under simultaneous time translation for all particles but not invariant if the times of each particle are translated individually. The delayed Rabinowitz action functional is invariant under individual time translation for each particle. Although its critical point equation looks like a Hamiltonian delay equation it is actually an ODE in disguise and nothing else than the critical point equation of the undelayed Rabinowitz action functional. We show that we can even interpolate between the two action functionals without changing the critical points and their actions. Moreover, for each of these interpolating action functionals we have compactness for gradient flow lines under a suitable restricted contact type assumption.