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 mapping f, the zeta-function is defined as the formal series

$\zeta _{f}(z)=\exp \left(\sum _{n=0}^{\infty }L(f^{n}){\frac {z^{n}}{n}}\right),$

where L(fⁿ) is the Lefschetz number of the nth iterate of 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.

Examples

For example, consider as space the unit circle, and let f be its reflection in the x-axis, or in other words θ → −θ. Then f has Lefschetz number 0, and f² is the identity map, which has Lefschetz number 2. Therefore we need

exp(2Σ t²ⁿ/2n)

which by considering

log (1 − t) + log (1 + t)

or otherwise is seen to be

1/(1 − t²).

A dull example: the identity map on X will have some power of 1/(1 − t) as its zeta-function. The power will be the sum of the Betti numbers of X.

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

Reference

• Alexander Fel’shtyn, Dynamical Zeta-Functions, Nielsen Theory and Reidemeister Torsion, arxiv, 1996