The study of Dynamical Zeta Functions is an active research area with lots of open problems. If you want to know more about them, below you will find the material needed to pay the entry fee and get up to speed.
In a nutshell, given a dynamical system there is a relationship between the spectrum of associated transfer operators and the poles of dynamical zeta functions via trace formulas. While a zeta function itself can be defined based solely on the dynamic of interest, rigorous statements on its properties are obtained through a careful analysis of (transfer) operators on (often anisotropic) Banach spaces tailored to the dynamic.
The surveys by V. Baladi '02 , M.Pollicott '10 and M. Pollicott '15 have the necessary introductory materials. If you want to follow the ideas of dynamical zeta functions in continuous time you can begin from this survey '11 . There is a 2018 book written by V. Baladi on Dynamical zeta functions for hyperbolic maps. Last, the lecture notes by F.Faure '18 will give you the necessary pointers to a family of techniques somehow disjoint from those presented in the previous notes.