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

is the Lefschetz number-th iterate.

Examples

has Lefschetz zeta function

is the Euler characteristic, i.e., the Lefschetz number of the identity map.

be the unit circle be reflection in the x 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