4 1. INTRODUCTION
More precisely, for δ(K)
0
, the zeta function FD
(K)
(s) has a pole s close to
s0
(n 1) ln κmax
(K)
2d(K)
+
π
d(K)
i ,
where the number s0 R is determined by the matrix B = {B(i, j)}i,j=1
p
defined
by B(i, j) = 1 if d(K) di,j (K) Γ0 (δ(K))γ0 and B(i, j) = 0 otherwise as follows.
In general the subshift σB : ΣB
+
−→ ΣB
+
of the shift σA : ΣA
+
−→ ΣA
+
(see below) is
not mixing, however there is a partition
ΣB
+
= X1 X2 . . . X
of ΣB
+
into compact and open subsets invariant under σB , so that the restriction of
σB onto each Xj is mixing (see Ch. 4). We then choose s0 R to be the maximal
number such that for some j = 1, . . . , the topological pressure of the function
(−s0f + ω)|Xj with respect to the shift σB : Xj −→ Xj is zero.
One should mention that the proof of this theorem in Ch. 7 below provides an
explicit estimate for the number
0
= 0(p, D0, d0, χ0, χ1, Γ0, γ0).
A special case when the gap condition (1.7) clearly holds is described in the
following.
Corollary 1.2. Let O1, O2, . . . , Op be points in
Rn
so that no three of them
lie on a line, and let D0 0 and χ0 1 be constants. There exists
0
0 such that
if K is an obstacle of the form (1.1) in
Rn,
where Ki are strictly convex disjoint
compact domains in Rn with C∞ boundaries such that Oi Ki and diam(Ki)
0
for all i = 1, . . . , p and K satisfies the conditions (1.2) and (1.3), then the MLPC
holds for K.
The latter is a generalization of Ikawa’s result mentioned above to finite unions
of strictly convex bodies of general shape (as long as they have bounded eccentric-
ity), while Theorem 1.1 is more general and much more difficult to prove.
In what folows we briefly describe Ikawa’s approach in dealing with the zeta
function ζ(s) and the extra difficulties we encounter in the present work.
For obstacles of the form (1.1) satisfying the no eclipse condition (H) there is
an obvious natural coding of the trapped billiard trajectories in ΩK using the shift
space
ΣA = = (ξm)m=−∞

: 1 ξi p, A(ξi, ξi+1) = 1 for all i } ,
where the p × p matrix A is defined by A(i, j) = 0 if i = j and A(i, j) = 1
otherwise. Namely, to any (x, u) ∂K × Sn−1 generating a billiard trajectory in
ΩK with infinitely many forward and backward reflections one assigns the sequence
ξ = (ξm)m=−∞ ΣA such that for any integer m the mth reflection point xm(ξ) of
the billiard trajectory γ(ξ) in ΩK generated by (x, u) belongs to ∂Kξm . Then the
shift map σA : ΣA −→ ΣA is conjugate to the billiard ball map on the set Λ of all
trapped points (x, u).
Ikawa [I3], [I4] showed that
ζ(s) = exp



k=1
1
k
σA(ξ)=ξk
e−s
ˆ
f
k
(ξ)+gk(ξ)+k π
i⎠

,
where hk(ξ) = h(ξ) + h(σAξ) + . . . +
h(σA−1ξ) k
for any function h on ΣA,
ˆ(ξ)
f =
x1(ξ) x0(ξ) , and g(ξ) is related to the principle curvatures of a convex front
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