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

$\zeta _{f}(t)=\prod _{i=0}^{n}\det(1-tf_{\ast }|H_{i}(X,{\mathbb {Q} }))^{(-1)^{i+1}}.$

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. {{cite arXiv}}: |class= ignored (help)