Zeta functions of dynamically tame Liouville domains

TL;DR

This paper introduces a new dynamical zeta function for a class of Liouville domains, extending its definition and invariance properties using symplectic homology, with applications to distinguishing certain open domains in R^4.

Contribution

It defines a dynamical zeta function for dynamically tame Liouville domains and proves its invariance under exact symplectomorphisms, expanding the understanding of symplectic invariants.

Findings

Zeta function invariance under symplectomorphisms

Extension of zeta function to dynamically tame domains

Examples of non-symplectomorphic domains close to a ball

Abstract

We define a dynamical zeta function for nondegenerate Liouville domains, in terms of Reeb dynamics on the boundary. We use filtered equivariant symplectic homology to (i) extend the definition of the zeta function to a more general class of "dynamically tame" Liouville domains, and (ii) show that the zeta function of a dynamically tame Liouville domain is invariant under exact symplectomorphism of the interior. As an application, we find examples of open domains in R^4, arbitrarily close to a ball, which are not symplectomorphic to open star-shaped toric domains.