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

+

= {ξ = (ξ0, ξ1, . . . , ξm, . . .) : 1 ≤ ξi ≤ p, A(ξi, ξi+1) = 1 for all i ≥ 0 } ,

and given θ ∈ (0, 1), define the metric dθ

+

on ΣA

+

by dθ

+

(ξ, η) = 0 if ξ = η and

dθ

+(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 Ω 0 we have the following:

(2.4) f(x) ≥ c0 (x ∈ ΣA)

+

, f

θ

≤ C0 ,

and

(2.5) ω

θ

≤ Ω , 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