Editing Artin–Mazur zeta function

In [[mathematics]], the '''Artin&ndash;Mazur [[zeta function]]''', named after [[Michael Artin]] and [[Barry Mazur]], is a function that is used for studying the [[iterated function]]s that occur in [[dynamical systems]] and [[fractals]].

It is defined from a given function <math>f</math> as the [[formal power series]]

:<math>\zeta_f(z)=\exp \left(\sum_{n=1}^\infty 
\bigl|\operatorname{Fix} (f^n)\bigr| \frac {z^n}{n}\right),</math>

where <math>\operatorname{Fix} (f^n)</math> is the set of [[Fixed point (mathematics)|fixed point]]s of the <math>n</math>th iterate of the function <math>f</math>, and <math>|\operatorname{Fix} (f^n)|</math> 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 <math>n</math>. This definition is formal in that the series does not always have a positive [[radius of convergence]].

The Artin&ndash;Mazur zeta function is invariant under [[topological conjugacy|topological conjugation]].

The [[Milnor–Thurston kneading theory|Milnor&ndash;Thurston theorem]] states that the Artin&ndash;Mazur zeta function of an interval map <math>f</math> is the inverse of the [[kneading determinant]] of <math>f</math>.

==Analogues==

The Artin&ndash;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&ndash;Mazur zeta function.

==See also==

*[[Lefschetz number]]
*[[Lefschetz zeta function|Lefschetz zeta-function]]

== References ==
* {{Citation | doi=10.2307/1970384 | last1=Artin | first1=Michael | author1-link=Michael Artin | last2=Mazur | first2=Barry | author2-link=Barry Mazur | title=On periodic points | mr=0176482 | year=1965 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=81 | pages=82–99 | issue=1 | publisher=Annals of Mathematics | jstor=1970384}}
* {{citation
 | last = Ruelle | first = David
 | issue = 8
 | journal = Notices of the American Mathematical Society
 | mr = 1920859
 | pages = 887–895
 | title = Dynamical zeta functions and transfer operators
 | url = https://www.ams.org/notices/200208/fea-ruelle.pdf
 | volume = 49
 | year = 2002}}
* {{citation | first1=Motoko |last1=Kotani | author1-link = Motoko Kotani| first2=Toshikazu | last2=Sunada |  author2-link=Toshikazu Sunada | title=Zeta functions of finite graphs | journal=J. Math. Sci. Univ. Tokyo | volume=7 | year=2000 | pages=7–25|citeseerx=10.1.1.531.9769 }}
* {{citation | title=Zeta Functions of Graphs: A Stroll through the Garden | volume=128 | series=Cambridge Studies in Advanced Mathematics | first=Audrey | last=Terras | author-link=Audrey Terras | publisher=[[Cambridge University Press]] | year=2010 | isbn=978-0-521-11367-0 | zbl=1206.05003 }}

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[[Category:Zeta and L-functions]]
[[Category:Dynamical systems]]
[[Category:Fixed points (mathematics)]]