Zeta functions of dynamically tame Liouville domains

We define a dynamical zeta function for nondegenerate Liouville domains, and also a more general class of "dynamically tame" Liouville domains. We use filtered equivariant symplectic homology to 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.