List of photonics equations

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

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

$\zeta (z)=\exp \left(\sum _{m\ge 1}\frac{z^m}{m}\sum _{x\in \mathrm{F}\mathrm{i}\mathrm{x}(f^m)}\mathrm{T}\mathrm{r}\left(\prod _{k=0}^{m-1}\varphi (f^k(x))\right)\right)$

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

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

which is the Artin–Mazur zeta function.

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

References

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