Artin–Mazur zeta function

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 [math]\displaystyle{ f }[/math] as the formal power series

[math]\displaystyle{ \zeta_f(z)=\exp \left(\sum_{n=1}^\infty \bigl|\operatorname{Fix} (f^n)\bigr| \frac {z^n}{n}\right), }[/math]

where [math]\displaystyle{ \operatorname{Fix} (f^n) }[/math] is the set of fixed points of the [math]\displaystyle{ n }[/math]th iterate of the function [math]\displaystyle{ f }[/math], and [math]\displaystyle{ |\operatorname{Fix} (f^n)| }[/math] 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 [math]\displaystyle{ n }[/math]. 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 [math]\displaystyle{ f }[/math] is the inverse of the kneading determinant of [math]\displaystyle{ f }[/math].

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 (Annals of Mathematics) 81 (1): 82–99, doi:10.2307/1970384, ISSN 0003-486X 
• Ruelle, David (2002), "Dynamical zeta functions and transfer operators", Notices of the American Mathematical Society 49 (8): 887–895, https://www.ams.org/notices/200208/fea-ruelle.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 978-0-521-11367-0