Topological and dynamical aspects of some spectral invariants of contact manifolds with circle action

TL;DR

This paper investigates spectral invariants like analytic torsion and eta invariants on CR contact manifolds with circle actions, linking their spectral properties to topological and dynamical aspects of the Reeb flow.

Contribution

It provides a new interpretation of spectral series as both topological invariants and dynamical functions of the Reeb flow on contact manifolds with circle symmetry.

Findings

Spectral series can be interpreted topologically.

Spectral series are dynamical functions of the Reeb flow.

Results apply to contact manifolds with circle actions.

Abstract

We study analytic torsion and eta like invariants on CR contact manifolds of any dimension admitting a circle transverse action, and equipped with a unitary representation. We show that, when defined using the spectrum of relevant operators arising in this geometry, the spectral series involved can been interpreted in their whole, both from a topological viewpoint, and as purely dynamical functions of the Reeb flow.