Artin-Mazur zeta function

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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

$\zeta_f(z)=\exp \sum_{n=1}^\infty \textrm{card} \left(\textrm{Fix} (f^n)\right) \frac {z^n}{n}$ ,

where $Fix(f n)$ is the set of fixed points of the n-th iterate of an iterated function f, and $\textrm{card} \left(\textrm{Fix} (f^n)\right)$ 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 invariant under topological conjugation.

The Milnor-Thurston theorem states that the Artin-Mazur zeta-function is the inverse of the kneading determinant of f.

[edit] 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.