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| In [[mathematics]], the '''[[Lefschetz]] [[zeta function|zeta-function]]''' is a tool used in topological periodic and [[fixed point (mathematics)|fixed point]] theory, and [[dynamical systems]]. Given a mapping ''f'', the zeta-function is defined as the formal series
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| :<math>\zeta_f(z) = \exp \left( \sum_{n=1}^\infty L(f^n) \frac{z^n}{n} \right), </math>
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| where ''L''(''f<sup>n</sup>'') is the [[Lefschetz number]] of the ''n''th [[iterated function|iterate]] of ''f''. This zeta-function is of note in topological periodic point theory because it is a single invariant containing information about all iterates of ''f''.
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| ==Examples==
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| The identity map on ''X'' has Lefschetz zeta function
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| :<math> \frac{1}{(1-t)^{\chi(X)}},</math>
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| where <math>\chi(X)</math> is the [[Euler characteristic]] of ''X'', i.e., the Lefschetz number of the identity map.
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| For a less trivial example, let ''X'' = '''S'''<sup>1</sup> (the [[unit circle]]), and let ''f'' be reflection in the ''x''-axis: or ''f''(θ) = −θ. Then ''f'' has Lefschetz number 2, and ''f''<sup>2</sup> is the identity map, which has Lefschetz number 0. All odd iterates have Lefschetz number 2, all even iterates have Lefschetz number 0. Therefore the zeta function of ''f'' is
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| :<math>\begin{align}
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| \zeta_f(t) & = \exp \left ( \sum_{n=1}^\infty \frac{2t^{2n+1}}{2n+1} \right ) \\
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| &=\exp \left ( \left \{2\sum_{n=1}^\infty \frac{t^n}{n}\right \} -\left \{2 \sum_{n=1}^\infty\frac{t^{2n}}{2n}\right \} \right ) \\
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| &=\exp \left(-2\log(1-t)+\log(1-t^2)\right)\\
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| &=\frac{1-t^2}{(1-t)^2} \\
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| &=\frac{1+t}{1-t}
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| \end{align}</math>
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| == Formula ==
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| If ''f'' is a continuous map on a compact manifold ''X'' of dimension ''n'' (or more generally any compact polyhedron), the zeta function is given by the formula
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| :<math>\zeta_f(t)=\prod_{i=0}^{n}\det(1-t f_\ast|H_i(X,\mathbf{Q}))^{(-1)^{i+1}}.</math>
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| Thus it is a rational function. The polynomials occurring in the numerator and denominator are essentially the characteristic polynomials of the map induced by ''f'' on the various homology spaces.
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| ==Connections ==
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| This generating function is essentially an [[algebra]]ic form of the [[Artin–Mazur zeta function|Artin–Mazur zeta-function]], which gives [[geometry|geometric]] information about the fixed and periodic points of ''f''.
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| ==See also==
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| *[[Lefschetz fixed point theorem]]
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| *[[Artin–Mazur zeta function|Artin–Mazur zeta-function]]
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| ==References==
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| *{{cite arXiv | title= Dynamical Zeta-Functions, Nielsen Theory and Reidemeister Torsion | year = 1996 | eprint=chao-dyn/9603017 | author1= Alexander Fel'shtyn | class= chao-dyn}}
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| [[Category:Zeta and L-functions]]
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| [[Category:Dynamical systems]]
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| [[Category:Fixed points (mathematics)]]
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