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[1]
\( {\displaystyle \zeta (z)=\exp \left(\sum _{m\geq 1}{\frac {z^{m}}{m}}\sum _{x\in \operatorname {Fix} (f^{m})}\operatorname {Tr} \left(\prod _{k=0}^{m-1}\phi (f^{k}(x))\right)\right)} \)
Examples
In the special case d = 1, φ = 1, we have[1]
\( {\displaystyle \zeta (z)=\exp \left(\sum _{m\geq 1}{\frac {z^{m}}{m}}\left|\operatorname {Fix} (f^{m})\right|\right)} \)
which is the Artin–Mazur zeta function.
The Ihara zeta function is an example of a Ruelle zeta function.[2]
See also
List of zeta functions
References
Terras (2010) p. 28
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.
Hellenica World - Scientific Library
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