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

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

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

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

which is the Artin–Mazur zeta function.

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

References

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