# Ruelle zeta function

In mathematics, the Ruelle zeta function is a zeta function associated with a dynamical system.

## Formal definition

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 is[1]

$\zeta(z) = \exp\left({ \sum_{m\ge1} \frac{z^m}{m} \sum_{x\in\mathrm{Fix}(f^m)} \mathrm{Tr} \left({ \prod_{k=0}^{m-1} \phi(f^k(x)) }\right) }\right)$

## Examples

In the special case d = 1, φ = 1, we have[1]

$\zeta(z) = \exp\left({ \sum_{m\ge1} \frac{z^m}{m} \left|{\mathrm{Fix}(f^m)}\right|}\right)$

which is the Artin–Mazur zeta function.

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