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) =
, s C ,
where γ runs over the set of periodic broken geodesics (billiard trajectories) in ΩK ,
is the period (length) of γ, the primitive period of γ, and 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
log ζ(s), where
ζ(s) = exp

(−1)mrγ em(−sTγ +δγ )
Here γ runs over the set of primitive periodic broken geodesics in Ω, = 0 if
γ has an even number of reflection points and = 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
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 so that if 0 0, then the MLPC holds
for K.
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