CHAPTER 2
An abstract meromorphicity theorem
Let A = (A(i, j))i,j=1
p
and B = (B(i, j))i,j=1
p
be p × p matrices consisting of 0’s
and 1’s such that B(i, j) = 1 implies A(i, j) = 1. Consider the symbol space
ΣA
+
= = (ξ0, ξ1, . . . , ξm, . . .) : 1 ξi p, A(ξi, ξi+1) = 1 for all i 0 } ,
and given θ (0, 1), define the metric
+
on ΣA
+
by
+
(ξ, η) = 0 if ξ = η and

+(ξ,
η) =
θk
if ξ = η, where k 0 is the maximal integer with ξi = ηi for
0 i k. Following [PP], for any function f : ΣA
+
−→ C set
varkf = sup{|f(ξ) f(η)| : ξi = ηi, 0 i k} , |f|θ = sup
varkf
θk
: k 0 ,
|f|∞ = sup{|f(ξ)| : ξ ΣA}
+
, f
θ
= |f|θ + |f|∞ .
Denote by Fθ(ΣA)
+
the space of complex functions f on ΣA
+
with f
θ
∞.
As in [I5], we will write i →B j if there exists a finite sequence i1 = i, i2, . . . , ik =
j such that B(ir, ir+1) = 1 for all r = 1, . . . , k 1. Relabeling the numbers 1, . . . , p
if necessary, we may assume that there exists an integer q such that 2 q p,
(2.1) if q i p , then B(i, j) = 0 for all j = 1, . . . , p ,
(2.2) i →B i for all i = 1, . . . , q ,
and
(2.3) i →B j implies j →B i for i, j = 1, . . . , q .
The Bernoulli shift σ : ΣA
+
−→ ΣA
+
is given by σ(ξ) = (ξ1, ξ2, . . .) for any
ξ = (ξ0, ξ1, ξ2, . . .) ΣA.
+
Given h Fθ(ΣA),
+
one defines hk Fθ(ΣA)
+
for any
k 1 by
hk(ξ) = h(ξ) + h(σ(ξ)) + . . . +
h(σk−1(ξ))
.
Assume that the real-valued function f Fθ(ΣA)
+
and ω Fθ(ΣA)
+
are such
that for some constants C0 1, c0 0 and 0 we have the following:
(2.4) f(x) c0 (x ΣA)
+
, f
θ
C0 ,
and
(2.5) ω
θ
, and ω(ξ) R whenever B(ξ0, ξ1) = 1 .
Denote by C = C(c0, C0, Ω) the family of pairs of functions (f, ω) satisfying the
above conditions.
Given (f, ω) C, 0 and Fθ(ΣA),
+
set Θ = (f, ω, ∆) for brevity and
define
u(Θ,)(ξ,
s) = −sf(ξ) + ω(ξ) + ∆(ξ) ln
7
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