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 as the formal power series

${\displaystyle \zeta _{f}(z)=\exp \left(\sum _{n=1}^{\infty }\operatorname {card} \left(\operatorname {Fix} (f^{n})\right){\frac {z^{n}}{n}}\right),}$

where Fix(ƒ ⁿ) is the set of fixed points of the nth iterate of the function ƒ, and card(Fix(ƒ ⁿ)) 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 n. 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 is the inverse of the kneading determinant of ƒ.