# 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 $\displaystyle{ f }$ as the formal power series

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

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

## 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.