Ruelle zeta function

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 isTerras (2010) p. 28

:$\backslash zeta(z)\; =\; \backslash exp\backslash left($

\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)

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

:$\backslash zeta(z)\; =\; \backslash exp\backslash left(\; \backslash sum\_\{m\backslash ge1\}\; \backslash frac\{z^m\}\; m\; \backslash left|\backslash operatorname\{Fix\}(f^m)\backslash right|\; \backslash right)$

which is the Artin–Mazur zeta function.

The Ihara zeta function is an example of a Ruelle zeta function.Terras (2010) p. 29

• List of zeta functions

{{reflist}}

• {{cite book | last1=Lapidus | first1=Michel L. | last2=van Frankenhuijsen | first2=Machiel | title=Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings | series=Springer Monographs in Mathematics | location=New York, NY | publisher=Springer-Verlag | year=2006 | isbn=0-387-33285-5 | zbl=1119.28005 }}
• {{cite journal | first1=Motoko |last1=Kotani | author1-link = Motoko Kotani | first2=Toshikazu | last2=Sunada | author2-link=Toshikazu Sunada |title=Zeta functions of finite graphs | journal=J. Math. Sci. Univ. Tokyo | volume=7 | year=2000 | pages=7–25 }}
• {{cite book | title=Zeta Functions of Graphs: A Stroll through the Garden | volume=128 | series=Cambridge Studies in Advanced Mathematics | first=Audrey | last=Terras | authorlink=Audrey Terras | publisher=Cambridge University Press | year=2010 | isbn=978-0-521-11367-0 | zbl=1206.05003 }}
• {{cite journal | first1=David |last1=Ruelle | author-link=David Ruelle | title=Dynamical Zeta Functions and Transfer Operators | url=https://www.ams.org/notices/200208/fea-ruelle.pdf | journal=Bulletin of AMS | volume=8 | issue=59 | year=2002 | pages=887–895 }}

Category:Zeta and L-functions