DYNAMICAL ZETA FUNCTIONS
5
The Artin-Mazur zeta function was adopted later by Milnor and Thurston
[67] to count periodic points for a piecewise monotone map of the interval.
The Artin-Mazur 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 Artin-Mazur and Lef-
schetz zeta functions, and D. Fried [40] defined twisted Artin-Mazur 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 Hasse-Weil zeta function.
Ruelle has found another generalization of the Artin-Mazur zeta function.
He was motivated by ideas from equilibrium statistical mechanics and has
replaced in the Artin-Mazur 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 Artin-Mazur zeta function. This symbolic dynamics is reminiscent
of the statistical mechanics of one-dimensional lattice spin system.
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