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

[math]\displaystyle{ \zeta(z) = \exp\left( \sum_{m\ge1} \frac{z^m}{m} \sum_{x\in\operatorname{Fix}(f^m)} \operatorname{Tr} \left( \prod_{k=0}^{m-1} \varphi(f^k(x)) \right) \right) }[/math]

Examples

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

[math]\displaystyle{ \zeta(z) = \exp\left( \sum_{m\ge1} \frac{z^m} m \left|\operatorname{Fix}(f^m)\right| \right) }[/math]

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

• Lapidus, Michel L.; van Frankenhuijsen, Machiel (2006). Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings. Springer Monographs in Mathematics. New York, NY: Springer-Verlag. ISBN 0-387-33285-5. 
• Kotani, Motoko; Sunada, Toshikazu (2000). "Zeta functions of finite graphs". J. Math. Sci. Univ. Tokyo 7: 7–25. 
• Terras, Audrey (2010). Zeta Functions of Graphs: A Stroll through the Garden. Cambridge Studies in Advanced Mathematics. 128. Cambridge University Press. ISBN 0-521-11367-9. 
• Ruelle, David (2002). "Dynamical Zeta Functions and Transfer Operators". Bulletin of AMS 8 (59): 887–895. https://www.ams.org/notices/200208/fea-ruelle.pdf.