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
,
where Fix(fn) is the set of fixed points of the n-th iterate of an iterated function f, and 
is the cardinality of this set of fixed points.
Note that the zeta-function is defined only if the 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.
[edit] See also
[edit] References
- M. Artin and Barry Mazur, On periodic points, Ann. of Math (2) 81 (1965) 82-99.
- David Ruelle, Dynamical Zeta Functions and Transfer Operators (2002) (PDF)
