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 continuous map $$f\colon X\to X$$, the zeta-function is defined as the formal series


 * $$\zeta_f(t) = \exp \left( \sum_{n=1}^\infty L(f^n) \frac{t^n}{n} \right), $$

where $$L(f^n)$$ is the Lefschetz number of the $$n$$-th 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 $$X = S^1$$ be the unit circle, and let $$f\colon S^1\to S^1$$ be reflection in the x-axis, that is, $$f(\theta) = -\theta$$. Then $$f$$ has Lefschetz number 2, while $$f^2$$ is the identity map, which has Lefschetz number 0. Likewise, all odd iterates have Lefschetz number 2, while 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.