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 formula_1, the zeta-function is defined as the formal series
where formula_3 is the Lefschetz number of the formula_4-th iterate of formula_5. This zeta-function is of note in topological periodic point theory because it is a single invariant containing information about all iterates of formula_5.
Examples.
The identity map on formula_7 has Lefschetz zeta function
where formula_9 is the Euler characteristic of formula_7, i.e., the Lefschetz number of the identity map.
For a less trivial example, let formula_11 be the unit circle, and let formula_12 be reflection in the "x"-axis, that is, formula_13. Then formula_5 has Lefschetz number 2, while formula_15 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 formula_5 is
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
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".