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(fn) 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

ζ ( z ) = exp ⁡ ( ∑ m ≥ 1 z m m ∑ x ∈ Fix ⁡ ( f m ) Tr ⁡ ( ∏ k = 0 m − 1 φ ( f k ( x ) ) ) ) {\displaystyle \zeta (z)=\exp \left(\sum _{m\geq 1}{\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)}

Examples

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

ζ ( z ) = exp ⁡ ( ∑ m ≥ 1 z m m | Fix ⁡ ( f m ) | ) {\displaystyle \zeta (z)=\exp \left(\sum _{m\geq 1}{\frac {z^{m}}{m}}\left|\operatorname {Fix} (f^{m})\right|\right)}

which is the Artin–Mazur zeta function.

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

See also

  • 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 .
  • 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. Vol. 128. Cambridge University Press. ISBN 978-0-521-11367-0. Zbl .
  • Ruelle, David (2002). (PDF). Bulletin of AMS. 8 (59): 887–895.