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
This zeta-function is of note in topological periodic point theory because it is a single invariant containing information about all iterates of
The identity map on
has Lefschetz zeta function where
, i.e., the Lefschetz number of the identity map.
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.
is 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.
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.