Английская Википедия: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 <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 points 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–Mazur zeta function is invariant under topological conjugation.
The Milnor–Thurston theorem states that the Artin–Mazur zeta function of an interval map <math>f</math> is the inverse of the kneading determinant of <math>f</math>.
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.
See also
References
