Lefschetz zeta function

${\displaystyle {\zeta }_f(t)=\exp \left(\sum _{n=1}^{\mathrm{\infty }}L(f^n)\frac{t^n}{n}\right),}$

where ${\displaystyle L(f^n)}$ is the Lefschetz number of the ${\displaystyle n}$-th iterate of ${\displaystyle f}$. This zeta-function is of note in topological periodic point theory because it is a single invariant containing information about all iterates of ${\displaystyle f}$.

Examples

The identity map on ${\displaystyle X}$ has Lefschetz zeta function

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

where ${\displaystyle \chi (X)}$ is the Euler characteristic of ${\displaystyle X}$, i.e., the Lefschetz number of the identity map.

For a less trivial example, let ${\displaystyle X=S^1}$ be the unit circle, and let ${\displaystyle f:S^1\to S^1}$ be reflection in the x-axis, that is, ${\displaystyle f(\theta )=-\theta }$. Then ${\displaystyle f}$ has Lefschetz number 2, while ${\displaystyle f^2}$ 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 ${\displaystyle f}$ is

${\displaystyle \begin{array}{cc}{\zeta }_f(t)& =\exp \left(\sum _{n=1}^{\mathrm{\infty }}\frac{2t^{2n+1}}{2n+1}\right)\\ & =\exp \left(\left\{2\sum _{n=1}^{\mathrm{\infty }}\frac{t^n}{n}\right\}-\left\{2\sum _{n=1}^{\mathrm{\infty }}\frac{t^{2n}}{2n}\right\}\right)\\ & =\exp \left(-2\log (1-t)+\log (1-t^2)\right)\\ & =\frac{1-t^2}{(1-t{)}^2}\\ & =\frac{1+t}{1-t}\end{array}}$