A variational approach to Givental's nonlinear Maslov index

TL;DR

This paper introduces a variational method using Rabinowitz Floer homology to define a homological count of discriminant points, leading to new obstructions for positive loops of contactomorphisms.

Contribution

It develops a novel variational approach to Givental's nonlinear Maslov index via Rabinowitz Floer homology, providing new tools for contact topology.

Findings

Defined a homological count of discriminant points

Established a Bott-Samelson type obstruction theorem

Linked growth rate to Givental's nonlinear Maslov index

Abstract

In this article we consider a variant of Rabinowitz Floer homology in order to define a homological count of discriminant points for paths of contactomorphisms. The growth rate of this count can be seen as an analogue of Givental's nonlinear Maslov index. As an application we prove a Bott-Samelson type obstruction theorem for positive loops of contactomorphisms.