Artin–Mazur zeta function explained

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

as the formal power series

where

is the set of fixed points of the th iterate of the function , and 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

. 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

is the inverse of the kneading determinant of .

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

• Lefschetz number
• Lefschetz zeta-function