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 = {ξ = (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