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 $f$ as the formal power series

$\zeta _{f}(z)=\exp \left(\sum _{n=1}^{\infty }{\bigl |}\operatorname {Fix} (f^{n}){\bigr |}{\frac {z^{n}}{n}}\right),$

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

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
• 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