Lefschetz zeta function

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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.

1 Examples
2 Connections
3 See also
4 References

[edit] 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 has Lefschetz zeta function

1/(1 − t)^χ(X),

where χ(X) is the Euler characteristic of X, i.e., the Lefschetz number of the identity map.

[edit] 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.