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minor sentence structure corrections

← Older revision		Revision as of 02:39, 30 January 2014
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	~~Hi there~~, ~~I am Sophia~~. ~~My husband doesn~~'~~t like~~ it the ~~way I do but what I really like doing~~ is ~~caving but I don~~'~~t have the time recently~~. ~~Credit authorising~~ is ~~how she tends~~ to ~~make~~ a ~~residing~~. ~~My wife and I live in Kentucky~~.~~<br><br>my weblog~~ ..~~. clairvoyants (~~[http://~~yoparo~~.~~org~~/~~content~~/~~hobby~~-~~one~~-~~thing~~-~~everyone~~-~~needs~~-~~have yoparo.org link web site~~])		In [[mathematics]], the '''Artin–Mazur [[zeta function]]''', named after [[Michael Artin]] and [[Barry Mazur]], is a tool for studying the [[iterated function]]s that occur in [[dynamical systems]] and [[fractals]].

			It is defined as the [[formal power series]]

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

			where Fix(''&fnof;''<sup> ''n''</sup>) is the set of [[Fixed point (mathematics)\|fixed point]]s of the ''n''th iterate of an [[iterated function]] ''&fnof;'', and card(Fix(''&fnof;''<sup> ''n''</sup>)) 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 kneading theory\|Milnor–Thurston theorem]] states that the Artin–Mazur zeta function is the inverse of the [[kneading determinant]] of ''&fnof;''.

			==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\|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\|Annals of Mathematics. Second Series]] \| issn=0003-486X \| volume=81 \| pages=82–99 \| issue=1 \| publisher=Annals of Mathematics \| jstor=1970384}}
			* [[David Ruelle]], [http://www.maths.ex.ac.uk/~mwatkins/zeta/ruelle.pdf Dynamical Zeta Functions and Transfer Operators] (2002) (PDF)
			* {{citation \| title=Zeta Functions of Graphs: A Stroll through the Garden \| volume=128 \| series=Cambridge Studies in Advanced Mathematics \| first=Audrey \| last=Terras \| authorlink=Audrey Terras \| publisher=[[Cambridge University Press]] \| year=2010 \| isbn=0-521-11367-9 \| zbl=1206.05003 }}

			{{DEFAULTSORT:Artin-Mazur zeta function}}
			[[Category:Zeta and L-functions]]
			[[Category:Dynamical systems]]
			[[Category:Fixed points (mathematics)]]

en>Xqbot: r2.7.3) (Robot: Adding da:Vandrende lavtryk

2012-07-17T18:03:26Z

r2.7.3) (Robot: Adding da:Vandrende lavtryk

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Hi there, I am Sophia. My husband doesn't like it the way I do but what I really like doing is caving but I don't have the time recently. Credit authorising is how she tends to make a residing. My wife and I live in Kentucky.<br><br>my weblog ... clairvoyants ([http://yoparo.org/content/hobby-one-thing-everyone-needs-have yoparo.org link web site])

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en>Chiswick Chap: /* top */ ce

en>Pccary: minor sentence structure corrections

en>Xqbot: r2.7.3) (Robot: Adding da:Vandrende lavtryk