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 ${\displaystyle f\colon X\to X}$, the zeta-function is defined as the formal series

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

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 ${\displaystyle f}$.