Ruelle zeta function

In mathematics, the Ruelle zeta function is a zeta function associated with a dynamical system. It is named after mathematical physicist David Ruelle.

Let f be a function defined on a manifold M, such that the set of fixed points Fix(f ⁿ) is finite for all n > 1. Further let φ be a function on M with values in d × d complex matrices. The zeta function of the first kind is¹

${\displaystyle \zeta (z)=\exp \left(\sum _{m\geq 1}{\frac {z^{m}}{m}}\sum _{x\in \operatorname {Fix} (f^{m})}\operatorname {Tr} \left(\prod _{k=0}^{m-1}\varphi (f^{k}(x))\right)\right)}$

In the special case d = 1, φ = 1, we have¹

${\displaystyle \zeta (z)=\exp \left(\sum _{m\geq 1}{\frac {z^{m}}{m}}\left|\operatorname {Fix} (f^{m})\right|\right)}$

which is the Artin–Mazur zeta function.

The Ihara zeta function is an example of a Ruelle zeta function.²