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 ${\displaystyle f}$.

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 ${\displaystyle f\colon S^{1}\to S^{1}}$ 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

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

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

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

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.

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.

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

• Fel'shtyn, Alexander (2000), "Dynamical zeta functions, Nielsen theory and Reidemeister torsion", Memoirs of the American Mathematical Society, 147 (699), arXiv:chao-dyn/9603017, MR 1697460