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 : X → X {\displaystyle f\colon X\to X} , the zeta-function is defined as the formal series
ζ f ( t ) = exp ( ∑ n = 1 ∞ L ( f n ) t n n ) , {\displaystyle \zeta _{f}(t)=\exp \left(\sum _{n=1}^{\infty }L(f^{n}){\frac {t^{n}}{n}}\right),}where L ( f n ) {\displaystyle L(f^{n})} is the Lefschetz number of the n {\displaystyle n} -th iterate of f {\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 f {\displaystyle f} .
Examples
The identity map on X {\displaystyle X} has Lefschetz zeta function
1 ( 1 − t ) χ ( X ) , {\displaystyle {\frac {1}{(1-t)^{\chi (X)}}},}where χ ( X ) {\displaystyle \chi (X)} is the Euler characteristic of X {\displaystyle X} , i.e., the Lefschetz number of the identity map.
For a less trivial example, let X = S 1 {\displaystyle X=S^{1}} be the unit circle, and let f : S 1 → S 1 {\displaystyle f\colon S^{1}\to S^{1}} be reflection in the x-axis, that is, f ( θ ) = − θ {\displaystyle f(\theta )=-\theta } . Then f {\displaystyle f} has Lefschetz number 2, while f 2 {\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 f {\displaystyle f} is
ζ f ( t ) = exp ( ∑ n = 1 ∞ 2 t 2 n + 1 2 n + 1 ) = exp ( { 2 ∑ n = 1 ∞ t n n } − { 2 ∑ n = 1 ∞ t 2 n 2 n } ) = exp ( − 2 log ( 1 − t ) + log ( 1 − t 2 ) ) = 1 − t 2 ( 1 − t ) 2 = 1 + t 1 − t {\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}}}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
ζ f ( t ) = ∏ i = 0 n det ( 1 − t f ∗ | H i ( X , Q ) ) ( − 1 ) i + 1 . {\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.
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
- 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