In mathematics, the Lefschetz zeta-function is a tool used in topological periodic and fixed point theory, and dynamical systems. Given a continuous map , the zeta-function is defined as the formal series where is the Lefschetz number of the -th iterate of . This zeta-function is of note in topological periodic point theory because it is a single invariant containing information about all iterates of . Examples The identity map on has Lefschetz zeta function where is the Euler characteristic of , i.e., the Lefschetz number of the identity map. For a less trivial example, let be the unit circle, and let be reflection in the x-axis, that is, . Then has Lefschetz number 2, while 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 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. See also Lefschetz fixed-point theorem Artin–Mazur zeta function Ruelle zeta function References Zeta and L-functions Dynamical systems Fixed points (mathematics)