CHAPTER 1
Introduction
Various zeta functions have been investigated in the fields of number theory,
geometry, dynamical systems and statistical physics. This work studies zeta func-
tions in a manner that follows the work of Artin and Mazur [1], Bowen and Lanford
[6], Ruelle [27] and Lind [18]. First, recall the zeta function that was defined by
Artin and Mazur.
Let φ : X −→ X be a homeomorphism of a compact space and Γn(φ) be the
number of fixed points of
φn.
The zeta function ζφ(s) for φ defined in [1] is
(1.1) ζφ(s) = exp

n=1
Γn (φ)
n
sn
.
Later, Bowen and Lanford [6] demonstrated that if φ is a shift of finite type,
then ζφ(s) is a rational function. In the simplest case, when a shift is generated by
a transition matrix A in Z, (1.1) is computed explicitly as
ζA(s) = exp


n=1
tr(An)
n
sn (1.2)
= (det(I sA))−1, (1.3)
and then
(1.4)
ζA(s) =
λ∈Σ(A)
(1
λs)−χ(λ),
where χ(λ) is a non-negative integer that is the algebraic multiplicity of eigenvalue
λ and Σ(A) is the spectrum of A. ζA(s) is a rational function that involves only
eigenvalues of A.
Lind [18] extended (1.1) to
Zd-actions
as follows. For a
Zd-action,
d 1, let
φ be an action of
Zd
on X. Denote the set of finite-index subgroups of
Zd
by Ld.
The zeta function ζφ, defined by Lind, is
(1.5) ζφ(s) = exp
L∈Ld
ΓL (φ)
[L]
s[L]
,
where [L] is the index
Zd/L
of L in
Zd
and ΓL(φ) is the number of fixed points
by
φn
for all n L. Lind [18] obtained some important results for ζφ, such as
conjugacy invariant and product formulae, and computed ζφ explicitly for some
interesting examples. Furthermore, he raised some fundamental problems for zeta
functions, including the following two.
Problem 7.2. [18] For ”finitely determined”
Zd-actions
φ such as shifts of
finite type, is there a reasonable finite description of ζφ(s)?
1
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