How to categorify dynamical zeta functions
by
Alexander Fel'shtyn
University of Szczecin and Boise State University

A programm of a categorification a la Khovanov of Weil type dynamical zeta functions is proposed.

(Categorification of the Weil zeta function) Let f: S→ S be a symplectomorphism of a compact surface. Then

Lf(z) : = exp æ è ∞ å n=1  L(fn) n zn ö ø = ∞ å n=0  L(Sd(f)) zd = ∞ å n=0  c(f, d) zd,

where L(fⁿ) is Lefschetz number, S^d(f): S^d(S) → S^d(S) is induced map on d-fold symmetric power of S and

c(f, d) = c(PFH(f, d)) = c(ECH(Tf, d)) = c(SWF(Tf, d))

is the Euler characteristic of the periodic Floer homology of degree d or the Euler characteristic of the embedded contact homology of degree d of the mapping torus T_f or the Euler characteristic of the corresponding Seiberg-Witten-Floer cohomology of degree d. There is a strong indication that SWF(T_f, d) homology give also a categorification of the Nielsen periodic point theory and corresponding minimal zeta function.

There are intriguing questions about categorifications of arithmetic L functions.

Date received: October 22, 2008