Artin–Mazur zeta function
It is defined as the formal power series
Note that the zeta function is defined only if the set of fixed points is finite for each n. This definition is formal in that the series does not always have a positive radius of convergence.
The Artin–Mazur zeta function is invariant under topological conjugation.
The Ihara zeta function of a graph can be interpreted as an example of the Artin–Mazur zeta function.
- Artin, Michael; Mazur, Barry (1965), "On periodic points", Annals of Mathematics, Second Series, Annals of Mathematics, 81 (1): 82–99, doi:10.2307/1970384, ISSN 0003-486X, JSTOR 1970384, MR 0176482
- David Ruelle, Dynamical Zeta Functions and Transfer Operators (2002) (PDF)
- 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