1. INTRODUCTION 5
at x0(ξ) determined by a specially defined phase function. (Then
e−2gk (ξ)
is the
product of the eigenvalues λj with |λj| 1 of the linear Poincar´ e map related to
the periodic billiard trajectory corresponding to ξ.)
Considering the case when the Ki’s are balls of radius and centres Pi, Ikawa
introduced a submatrix B of A so that B(i, j) = 1 iff |PiPj| = max and B(i, j) = 0
otherwise, and showed that ζ(s) = Z(s c( )) for some constant c( ) C, where
Z(s) = exp



m=1
1
m
σA m(ξ)=ξ
e−s
ˆ
f m(ξ)+ˆm(ξ)+∆m ω (ξ) ln


,
ˆ(ξ) ω is an appropriately defined function (depending on K), ∆(ξ) = 0 if B(ξ0, ξ1) =
1 and ∆(ξ) 0 otherwise. He then proved that there exist s0 R and δ 0 such
that if is sufficiently small, then Z(s) is meromorphic in = {s C : |s−s0| δ}
with a pole s in such that s s0 as 0. This implies that for such , ζ(s)
has a meromorphic continuation in a disk + c( ) close to the line of absolute
convergence with a pole in the same disk, so the MLPC holds for K.
To study Z(s), Ikawa compared it with a zeta function of the form
(1.8) Z0(s) = exp



m=1
1
m
σm(ξ)=ξ
A
e−sfm(ξ)+ωm(ξ)⎠

for some (much simpler) functions f and ω determined by the points Pi. His study
of phase functions and propagation of convex fronts in ΩK under the action of the
billiard flow ([I1], [I2]) was then employed to show that
ˆ
f f and ˆ ω ω as 0
with respect to an appropriate norm.
Using a well-known lemma of Sinai [Si1], one can consider the functions f, ω,
ˆ,
f ˆ, ω etc. as functions on ΣA.
+
One can then use transfer (Ruelle) operators to
study the zeta functions Z(s) and Z0(s). Using this kind of tools, Ikawa proved
an ‘abstract’ meromorphicity theorem (cf. e.g. Theorem 1 in [I5]) which claims
that for certain pairs (f, ω) of functions on ΣA
+
there exist s0 R and δ 0 having
the properties described above for any
ˆ,
f ˆ ω and satisfying certain assumptions
so that
ˆ
f and ˆ ω are sufficiently close to f and ω, respectively. This is the core of
Ikawa’s method.
To prove his abstract meromorphicity theorem, Ikawa needed to consider a
modified transfer operator
˜
L
−sf+ω
acting as 0 on a significant part of ΣA.
+
The
‘essential part’ of
˜
L
−sf+ω
decomposes into a direct sum of standard transfer opera-
tors acting on symbolic spaces ΣCj
+
with irreducible (but in general not apperiodic)
matrices Cj , and the classical Ruelle-Perron-Frobenius theorem can be applied to
the restriction of
˜
L
−sf+ω
to each ΣCj
+
. It turns out that
˜
L
−sf+ω
is quasi-compact
(its point spectrum is the union of the point spectra of its restrictions to the sub-
spaces ΣCj
+
). Choosing s0 R appropriately, 1 is an isolated (possibly multiple)
eigenvalue of
˜
L
−sf+ω
and the rest of the spectrum lies in {z C : |z|≤ 1}. It has
been known since results of Ruelle (1976) and Parry (1984) (cf. e.g. Ch. 5 in [PP])
that in this situation the weighted dynamical zeta function (defined in a similar
way to Z0(s)) is meromorphic in a neighbourhood of s0 with a single pole at
s0. Using a similar more general result of Pollicott [Po2] (see also Haydn [H]) and
basic facts from perturbation theory of linear operators, Ikawa succeeded to derive
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