Ruelle zeta function

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

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^[1]

$\zeta(z) = \exp\left({ \sum_{m\ge1} \frac{z^m}{m} \sum_{x\in\mathrm{Fix}(f^m)} \mathrm{Tr} \left({ \prod_{k=0}^{m-1} \phi(f^k(x)) }\right) }\right)$

Examples

In the special case d = 1, φ = 1, we have^[1]

$\zeta(z) = \exp\left({ \sum_{m\ge1} \frac{z^m}{m} \left|{\mathrm{Fix}(f^m)}\right|}\right)$

which is the Artin–Mazur zeta function.

The Ihara zeta function is an example of a Ruelle zeta function.^[2]

See also

• List of zeta functions

References

1. Terras (2010) p. 28
2. Terras (2010) p. 29

• 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. Zbl 1119.28005.
• 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. Zbl 1206.05003.