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:$zeta\_f(z)\; =\; exp\; left(\; sum\_\{n=0\}^infty\; L(f^n)\; frac\{z^n\}\{n\}\; ight),$where "L(fⁿ)" is the Lefschetz number of the "n"th 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 "f"² is the identity map, which has Lefschetz number 2. Therefore we need

:exp(2Σ "t"^2"n"/2"n")

which by considering

:log (1 − "t") + log (1 + "t")

or otherwise is seen to be

:1/(1 − "t"²).

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.

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".

ee also

*Lefschetz fixed point theorem
*Artin-Mazur zeta-function

References

*cite arXiv | author=Felshtyn, A. | title= Dynamical Zeta-Functions, Nielsen Theory and Reidemeister Torsion | year = 1996 | eprint=chao-dyn/9603017