Difference between revisions of "Artin-Mazur zeta function"
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| − | Let $\mathrm{Fix}(f^n)$ be the set of [[fixed points]] of the $n$th [[iterate]] $f^n$ of $f$. Let $\mathrm{Card}(Fix)(f^n)$ denote the [[cardinality]] of the set $\mathrm{Fix}(f^n)$. The Artin-Mazur zeta function is | + | Let $\mathrm{Fix}(f^n)$ be the set of [[fixed points]] of the $n$th [[iterate]] $f^n$ of $f$. Let $\mathrm{Card}(\mathrm{Fix})(f^n)$ denote the [[cardinality]] of the set $\mathrm{Fix}(f^n)$. The Artin-Mazur zeta function is |
$$\zeta_f(z)=\exp \left( \displaystyle\sum_{k=1}^{\infty} \mathrm{Card}(\mathrm{Fix})[f^n] \dfrac{z^n}{n} \right).$$ | $$\zeta_f(z)=\exp \left( \displaystyle\sum_{k=1}^{\infty} \mathrm{Card}(\mathrm{Fix})[f^n] \dfrac{z^n}{n} \right).$$ | ||
=References= | =References= | ||
[http://www.ihes.fr/~ruelle/PUBLICATIONS/%5B105%5D.pdf] | [http://www.ihes.fr/~ruelle/PUBLICATIONS/%5B105%5D.pdf] | ||
Revision as of 04:04, 12 April 2015
Let $\mathrm{Fix}(f^n)$ be the set of fixed points of the $n$th iterate $f^n$ of $f$. Let $\mathrm{Card}(\mathrm{Fix})(f^n)$ denote the cardinality of the set $\mathrm{Fix}(f^n)$. The Artin-Mazur zeta function is $$\zeta_f(z)=\exp \left( \displaystyle\sum_{k=1}^{\infty} \mathrm{Card}(\mathrm{Fix})[f^n] \dfrac{z^n}{n} \right).$$
