DYNAMICAL ZETA FUNCTIONS
5
The ArtinMazur zeta function was adopted later by Milnor and Thurston
[67] to count periodic points for a piecewise monotone map of the interval.
The ArtinMazur zeta function was historically the first dynamical zeta
function for discrete dynamical system. The next dynamical zeta function
was defined by Smale [87] .This is the Lefschetz zeta function of discrete
dynamical system:
Lf(z) := ex p (£ ^ A
,
\n=l
U
J
where
dimX
L(fn) := £ (!) f c ^ [/* •• Hk(X;Q)  Hk{X;dj\
k=0
is the Lefschetz number of fn. Smale considered Lf(z) in the case when
/ is diffeomorphism of a compact manifold, but it is well defined for any
continuous map / of compact polyhedron X. The Lefschetz zeta function is
a rational function of z and is given by the formula:
dimX
Lf(z)= n det(//,
fc
.z)
(

1)+1
.
fc=0
Afterwards, J.Franks [38] defined reduced mod 2 ArtinMazur and Lef
schetz zeta functions, and D. Fried [40] defined twisted ArtinMazur and
Lefschetz zeta functions, which have coefficients in the group rings 2ZH or
/Z2H of an abelian group H. The above zeta functions are directly analogous
to the HasseWeil zeta function.
Ruelle has found another generalization of the ArtinMazur zeta function.
He was motivated by ideas from equilibrium statistical mechanics and has
replaced in the ArtinMazur zeta function [82] simple counting of the periodic
points by counting with weights. He defined the Ruelle zeta function as
(E
oo 7n ra—1
 E n *(/*(*))
n=l
n
xGFix
(fn)
k=0
where g : X — • (Fis a weight function(if g = 1 we recover Ff(z)).
Dynamical zeta functions have relations with statistical mechanics( en
tropy, pressure, Gibbs states, equilibrium states). Manning used Markov
partitions and corresponding symbolic dynamics in his proof of the rational
ity of the ArtinMazur zeta function. This symbolic dynamics is reminiscent
of the statistical mechanics of onedimensional lattice spin system.