4
ALEXANDER FEL'SHTYN
Note that £(z, V) can be written as a Euler product
over all primitive periodic orbits 7 of F on V. For comparison with Riemann's
zeta function one has to put z =
q~s.
Certain conjectures proposed by
Weil [98] on the properties of £(z, V) led to lot of work by Weil, Dwork,
Grothendieck, and complete proof was finally obtained by Deligne [15]. The
story of the Weil conjectures is one of the most striking instances exhibiting
the fundamental unity of mathematics. It is found that £(2, V") is a rational
function of z with a functional equation :
2ra
c(*,vo =
nw(~1),+1
i=0
where the zeros of the polynomial Pi have absolute value
q~lj/2
and the Pi(z)
have a cohomological interpretation: the polynomial Pi is roughly the char-
acteristic polynomial associated with the induced action of the Frobenius
morphism on the etale cohomology: Pi(z) = det(l z
F*\Hl(V)).
0.1.5 Dynamical zeta functions
Inspired by the Hasse-Weil zeta function of an algebraic variety over a finite
field, Artin and Mazur [5] defined the Artin - Mazur zeta function for an
arbitrary map / : X X of a topological space X:
where
F(fn)
is the number of isolated fixed points of
fn.
Artin and Mazur
showed that for a dense set of the space of smooth maps of a compact smooth
manifold into itself the Artin-Mazur zeta function Ff(z) has a positive ra-
dius of convergence.Later Manning [64] proved the rationality of the Artin -
Mazur zeta function for diffeomorphisms of a smooth compact manifold sat-
isfying Smale axiom A, after partial results were obtained by Williams and
Guckenheimer. On the other hand there exist maps for which Artin-Mazur
zeta function is transcendental [11].
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