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 Dδ = {s ∈ C : |s−s0| δ}

with a pole s in Dδ such that s → s0 as → 0. This implies that for such , ζ(s)

has a meromorphic continuation in a disk Dδ + 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 Dδ 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