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Lefschetz zeta function
In mathematics, the Lefschetz zeta-function is a tool used in topological periodic and fixed point theory, and dynamical systems. Given a mapping ƒ, the zeta-function is defined as the formal series
\( \zeta_f(z) = \exp \left( \sum_{n=1}^\infty L(f^n) \frac{z^n}{n} \right), \)
where L(ƒ n) is the Lefschetz number of the nth iterate of ƒ. This zeta-function is of note in topological periodic point theory because it is a single invariant containing information about all iterates of ƒ.
Examples
The identity map on X has Lefschetz zeta function
\( 1/(1 − t)χ(X), \)
where χ(X) is the Euler characteristic of X, i.e., the Lefschetz number of the identity map.
For a less trivial example, consider as space the unit circle, and let ƒ be its reflection in the x-axis, or in other words θ → −θ. Then ƒ has Lefschetz number 2, and ƒ2 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 ƒ is
\( \zeta_f(t)=\exp \sum_{n=1}^\infty \frac{2t^{2n+1}}{2n+1}. \)
Via intermediate expressions
\( =\exp 2\Big(\sum_{n=1}^\infty \frac{t^n}{n}-\sum_{n=1}^\infty\frac{t^{2n}}{2n}\Big)=\exp \Big( -2\log(1-t)+\log(1-t^2)\Big)=\frac{1-t^2}{(1-t)^2}. \)
This is seen to be equal to
\( \zeta_f(t)=\frac{1+t}{1-t}. \)
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
\( \zeta_f(t)=\prod_{i=0}^{n}\det(1-t f_\ast|H_i(X,{\Bbb Q}))^{(-1)^{i+1}}. \)
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 ƒ.
See also
Lefschetz fixed point theorem
Artin–Mazur zeta-function
References
Alexander Fel'shtyn (1996). "Dynamical Zeta-Functions, Nielsen Theory and Reidemeister Torsion". arXiv:chao-dyn/9603017 [chao-dyn].
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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