# Artin-Mazur zeta function

### From Thikipedia

In maths, the **Artin-Mazur zeta-function** is a tool for studying the iterated functions that occur in dynamical systems and fractals.

It is defined as the formal power series

**Failed to parse (Can't write to or create math temp directory): \\zeta_f(z)=\\exp \\sum_{n=1}^\\infty \\textrm{card} \\left(\\textrm{Fix} (f^n)\\right) \\frac {z^n}{n}**

,
where **Failed to parse (Can't write to or create math temp directory): \\textrm{Fix}(f^n)**

is the set of fixed points of then-th iterate of an iterated functionf, andFailed to parse (Can't write to or create math temp directory): \\textrm{card} \\left(\\textrm{Fix} (f^n)\\right)is the cardinality of this set of fixed points.

Note that the zeta-function is defined only if the set of fixed points is finite. This definition is formal in that it does not always have a positive radius of convergence.

The Artin-Mazur zeta-function is not invariant under topological conjugation.

The Milnor-Thurston theorem states that the Artin-Mazur zeta-function is the inverse of the kneading determinant of *f*.

The Artin-Mazur zeta-function is equivalent to the Weil zeta-function when there is a diffeomorphism on a compact manifold.

Under certain cases, the Artin-Mazur zeta-function can be related to the Ihara zeta-function of a graph.

## [edit] See also

## [edit] References

- M. Artin and B. Mazur,
*On periodic points*, Ann. of Math (2) 81 (1965) 82-99. - David Ruelle, Dynamical Zeta Functions and Transfer Operators (2002) (PDF)