Editing Artin–Mazur zeta function (section)

In [[mathematics]], the '''Artin&ndash;Mazur [[zeta function]]''', named after [[Michael Artin]] and [[Barry Mazur]], is a function that is used for studying the [[iterated function]]s that occur in [[dynamical systems]] and [[fractals]].

It is defined from a given function <math>f</math> as the [[formal power series]]

:<math>\zeta_f(z)=\exp \left(\sum_{n=1}^\infty 
\bigl|\operatorname{Fix} (f^n)\bigr| \frac {z^n}{n}\right),</math>

where <math>\operatorname{Fix} (f^n)</math> is the set of [[Fixed point (mathematics)|fixed point]]s of the <math>n</math>th iterate of the function <math>f</math>, and <math>|\operatorname{Fix} (f^n)|</math> 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 <math>n</math>. This definition is formal in that the series does not always have a positive [[radius of convergence]].

The Artin&ndash;Mazur zeta function is invariant under [[topological conjugacy|topological conjugation]].

The [[Milnor–Thurston kneading theory|Milnor&ndash;Thurston theorem]] states that the Artin&ndash;Mazur zeta function of an interval map <math>f</math> is the inverse of the [[kneading determinant]] of <math>f</math>.