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 f, 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(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

$\frac{1}{(1-t)^{\chi(X)}},$

where $\chi(X)$ 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

$\begin{align} \zeta_f(t) & = \exp \left( \sum_{n=1}^\infty \frac{2t^{2n+1}}{2n+1} \right) \\ &=\exp \left( \left\{2\sum_{n=1}^\infty \frac{t^n}{n}\right\} -\left \{2 \sum_{n=1}^\infty\frac{t^{2n}}{2n}\right\} \right) \\ &=\exp \left(-2\log(1-t)+\log(1-t^2)\right)\\ &=\frac{1-t^2}{(1-t)^2} \\ &=\frac{1+t}{1-t} \end{align}$

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

$\zeta_f(t)=\prod_{i=0}^{n}\det(1-t f_\ast|H_i(X,\mathbf{Q}))^{(-1)^{i+1}}.$

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 f.

See also

• Lefschetz fixed point theorem
• Artin–Mazur zeta-function