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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 | |||
:<math>\zeta_f(z) = \exp \left( \sum_{n=1}^\infty L(f^n) \frac{z^n}{n} \right), </math> | |||
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''. | |||
==Examples== | |||
The identity map on ''X'' has Lefschetz zeta function | |||
:<math> \frac{1}{(1-t)^{\chi(X)}},</math> | |||
where <math>\chi(X)</math> is the [[Euler characteristic]] of ''X'', i.e., the Lefschetz number of the identity map. | |||
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 | |||
:<math>\begin{align} | |||
\zeta_f(t) & = \exp \left ( \sum_{n=1}^\infty \frac{2t^{2n+1}}{2n+1} \right ) \\ | |||
&=\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 ) \\ | |||
&=\exp \left(-2\log(1-t)+\log(1-t^2)\right)\\ | |||
&=\frac{1-t^2}{(1-t)^2} \\ | |||
&=\frac{1+t}{1-t} | |||
\end{align}</math> | |||
== Formula == | |||
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 | |||
:<math>\zeta_f(t)=\prod_{i=0}^{n}\det(1-t f_\ast|H_i(X,\mathbf{Q}))^{(-1)^{i+1}}.</math> | |||
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. | |||
==Connections == | |||
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''. | |||
==See also== | |||
*[[Lefschetz fixed point theorem]] | |||
*[[Artin–Mazur zeta function|Artin–Mazur zeta-function]] | |||
==References== | |||
*{{cite arXiv | title= Dynamical Zeta-Functions, Nielsen Theory and Reidemeister Torsion | year = 1996 | eprint=chao-dyn/9603017 | author1= Alexander Fel'shtyn | class= chao-dyn}} | |||
[[Category:Zeta and L-functions]] | |||
[[Category:Dynamical systems]] | |||
[[Category:Fixed points (mathematics)]] | |||
Revision as of 15:14, 26 February 2013
In mathematics, the Lefschetz zeta-function is a tool used in topological periodic and fixed point theory, and dynamical systems. Given a mapping f, the zeta-function is defined as the formal series
where L(fn) is the Lefschetz number of the nth 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.
Examples
The identity map on X has Lefschetz zeta function
where is the Euler characteristic of X, i.e., the Lefschetz number of the identity map.
For a less trivial example, let X = S1 (the unit circle), and let f be reflection in the x-axis: or f(θ) = −θ. Then f has Lefschetz number 2, and f2 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
Formula
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
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.
Connections
This generating function is essentially an algebraic form of the Artin–Mazur zeta-function, which gives geometric information about the fixed and periodic points of f.