Editing Artin-Mazur zeta function

In [[mathematics]], the '''Artin-Mazur [[zeta function|zeta-function]]''' is a tool for studying the [[iterated function]]s that occur in [[dynamical systems]] and [[fractals]].

It is defined as the [[formal power series]] 
:<math>\\zeta_f(z)=\\exp \\sum_{n=1}^\\infty \\textrm{card} 
\\left(\\textrm{Fix} (f^n)\\right) \\frac {z^n}{n}</math>,
where <math>\\textrm{Fix}(f^n)</math> is the set of [[fixed point]]s of the ''n''-th iterate of an [[iterated function]] ''f'', and <math>\\textrm{card} 
\\left(\\textrm{Fix} (f^n)\\right)</math> is the [[cardinality]] of this set of fixed points.

Note that the zeta-function is defined only if 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|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|Ihara zeta-function]] of a graph. 

==See also==

*[[Lefschetz number]]
*[[Lefschetz zeta function|Lefschtetz zeta-function]]

== References ==  

* [[M. Artin]] and [[B. 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)

[[Category:Zeta and L-functions]]
[[Category:Dynamical systems]]
[[Category:Fixed points]]