Artin–Mazur zeta function

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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]\displaystyle{ f }[/math] as the formal power series

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

where [math]\displaystyle{ \operatorname{Fix} (f^n) }[/math] is the set of fixed points of the [math]\displaystyle{ n }[/math]th iterate of the function [math]\displaystyle{ f }[/math], and [math]\displaystyle{ |\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]\displaystyle{ 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]\displaystyle{ f }[/math] is the inverse of the kneading determinant of [math]\displaystyle{ f }[/math].


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