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 ${\displaystyle f}$ as the formal power series

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

where ${\displaystyle \operatorname {Fix} (f^{n})}$ is the set of fixed points of the ${\displaystyle n}$th iterate of the function ${\displaystyle f}$, and ${\displaystyle |\operatorname {Fix} (f^{n})|}$ 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 ${\displaystyle n}$. 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 of an interval map ${\displaystyle f}$ is the inverse of the kneading determinant of ${\displaystyle f}$.