CHAPTER 2 An abstract meromorphicity theorem Let A = (A(i, j))p i,j=1 and B = (B(i, j))p i,j=1 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 d+ θ on ΣA + by d+(ξ, θ η) = 0 if ξ = η and d+(ξ, θ 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 : Σ+ A −→ 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 : Σ+ A −→ Σ+ A is given by σ(ξ) = (ξ1, ξ2, . . .) for any ξ = (ξ0, ξ1, ξ2, . . .) Σ+. A 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 Σ+) A , 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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