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.

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

$\zeta (z)=\exp (\sum _{m\ge 1}\frac{z^m}{m}\sum _{x\in \mathrm{Fix}(f^m)}\mathrm{Tr}(\prod _{k=0}^{m-1}\phi (f^k(x))))$

Examples

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

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

which is the Artin–Mazur zeta function.

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

See also

• List of zeta functions

References

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