2 1. INTRODUCTION

Melrose and the survey articles of Sj¨ ostrand [Sj2], Zworski [Z1], [Z2], and Vodev

[V] and the references there for a comprehensive information in this direction.

See also the papers of Tang and Zworski [TZ] and Stefanov [Ste]. Some of these

results have consequences concerning the MLPC. One particular result of this kind

was obtained by Stefanov and Vodev [SteV] as an application of their study of

resonances based on Popov’s [P] construction of quasimodes. Namely it is shown

in [SteV] that if there exists an elliptic periodic trajectory in ΩK satisfying some

non-degeneracy conditions, then there is a sequence of resonances converging to the

real axis; in particular the MLPC holds.

In this paper we deal with the case when K has the form

(1.1) K = K1 ∪ K2 ∪ . . . ∪ Kp ,

where p ≥ 3 and Ki are strictly convex disjoint compact domains in Rn with C∞

boundaries satisfying the following no eclipse condition introduced by Ikawa:

(H) Kk ∩ convex hull(Ki ∪ Kj ) = ∅ for all k = i = j = k .

To deal with the MLPC for obstacles of the form (1.1), Ikawa [I3] introduced

the zeta function

FD(s) =

γ∈Ξ

(−1)mγ

Tγ |I − Pγ

|−1/2e−sdγ

, s ∈ C ,

where γ runs over the set of periodic broken geodesics (billiard trajectories) in ΩK ,

dγ is the period (length) of γ, Tγ the primitive period of γ, and Pγ the linear

Poincar´ e map associated to γ. He then showed that existence of analytic singu-

larities of FD(s) implies existence of a band 0 Im(z) α containing an infinite

number of scattering poles λj , i.e. the MLPC holds in such cases.

Clearly FD(s) is a Dirichlet series. Let z0 be its abscissa of absolute conver-

gence. Ikawa showed (in the case n = 3) that there exists α 0 such that in the

region z0 − α Re(s) ≤ z0 the analytic singularities of FD(s) coincide with these

of

d

ds

log ζ(s), where

ζ(s) = exp

∞

m=0

γ

(−1)mrγ em(−sTγ +δγ )

.

Here γ runs over the set of primitive periodic broken geodesics in Ω, rγ = 0 if

γ has an even number of reflection points and rγ = 1 otherwise, and δγ ∈ R is

determined by the spectrum of the linear Poincar´ e map related to γ. The function

ζ(s) is rather similar to a dynamically defined zeta function (see below). Ikawa

[I4], [I5] succeeded to implement results of W. Parry, M. Pollicott and N. Haydn

concerning the spectrum of the Ruelle operator and obtained a suﬃcient condition

for ζ(s) (and therefore FD(s)) to have a pole in a small neighbourhood of z0 in C.

From this he derived:

Ikawa [I5]: Let O1, O2, . . . , Op be points in

R3

so that no three of them lie on a

line, and let K be the union of the balls with centers Oi (i = 1, . . . , p) and the same

radius 0. Then there exists

0

0 so that if 0 ≤ 0, then the MLPC holds

for K.