In mathematics, the Artin–Mazur zeta function, named after Michael Artin and Barry Mazur, is a function that is used for studying the iterated functions that occur in dynamical systems and fractals. It is defined from a given function f {\displaystyle f} as the formal power series
ζ f ( z ) = exp ( ∑ n = 1 ∞ | Fix ( f n ) | z n n ) , {\displaystyle \zeta _{f}(z)=\exp \left(\sum _{n=1}^{\infty }{\bigl |}\operatorname {Fix} (f^{n}){\bigr |}{\frac {z^{n}}{n}}\right),}
where Fix ( f n ) {\displaystyle \operatorname {Fix} (f^{n})} is the set of fixed points of the n {\displaystyle n} th iterate of the function f {\displaystyle f} , and | Fix ( f n ) | {\displaystyle |\operatorname {Fix} (f^{n})|} is the number of fixed points (i.e. the cardinality of that set). Note that the zeta function is defined only if the set of fixed points is finite for each n {\displaystyle 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 Milnor–Thurston theorem states that the Artin–Mazur zeta function of an interval map f {\displaystyle f} is the inverse of the kneading determinant of f {\displaystyle f} .
Analogues The Artin–Mazur zeta function is formally similar to the local zeta function, when a diffeomorphism on a compact manifold replaces the Frobenius mapping for an algebraic variety over a finite field. The Ihara zeta function of a graph can be interpreted as an example of the Artin–Mazur zeta function.
See also Lefschetz number Lefschetz zeta-function
References Artin, Michael; Mazur, Barry (1965), "On periodic points", Annals of Mathematics, Second Series, 81 (1), Annals of Mathematics: 82–99, doi:10.2307/1970384, ISSN 0003-486X, JSTOR 1970384, MR 0176482 Ruelle, David (2002), "Dynamical zeta functions and transfer operators" (PDF), Notices of the American Mathematical Society, 49 (8): 887–895, MR 1920859 Kotani, Motoko; Sunada, Toshikazu (2000), "Zeta functions of finite graphs", J. Math. Sci. Univ. Tokyo, 7: 7–25, CiteSeerX 10.1.1.531.9769 {{citation}}: Cite uses deprecated parameter |citeseerx= (help) 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 1206.05003