Artin-Mazur zeta function

In mathematics, 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 :$zeta\_f(z)=exp\; sum\_\{n=1\}^infty\; extrm\{card\}\; left(\; extrm\{Fix\}\; (f^n)\; ight)\; frac\; \{z^n\}\{n\}$,where $extrm\{Fix\}(f^n)$ is the set of fixed points of the "n"-th iterate of an iterated function "f", and $extrm\{card\}\; left(\; extrm\{Fix\}\; (f^n)\; ight)$ 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 invariant under topological conjugation.

The Milnor-Thurston theorem states that the Artin-Mazur zeta-function 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.

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

ee also

*Lefschetz number
*Lefschetz zeta-function

References

* M. Artin and Barry Mazur, "On periodic points", Ann. of Math (2) 81 (1965) 82-99.
* David Ruelle, [http://www.maths.ex.ac.uk/~mwatkins/zeta/ruelle.pdf Dynamical Zeta Functions and Transfer Operators] (2002) (PDF)