Ruelle zeta function

Summary

The Ruelle zeta function of a map is a weighted count of its periodic points. Each fixed point of $f^n$ is counted with the product of a weight function along its orbit. With the weight $1$ it is the Artin–Mazur zeta function. It is a basic tool of the thermodynamic formalism, where its poles and zeros are related to the eigenvalues of transfer operators. Ruelle also defined a version for flows, with a product over closed orbits. For geodesic flows of hyperbolic surfaces, this version is close to the Selberg zeta function.