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<sup> n</sup>) is finite for all n > 1. Further let ÃÂ be a function on M with values in d ÃÂ d complex matrices. The zeta function of the first kind is
In the special case d = 1, ÃÂ = 1, we have
which is the ArtinâÂÂMazur zeta function.
The Ihara zeta function is an example of a Ruelle zeta function.