10 3. PRELIMINARIES λ, and the essential spectral radius of Lf is θ λ. The eigenfunction h satisfies (3.2) h θ ≤ 6 qMτ τ b θ2τ (1 − θτ ) e4τ b/(1−θτ ) e2Mτ|f|∞ and (3.3) min h ≥ 1 e2 τ b/(1−θτ ) qMτ e2Mτ|f|∞ . Moreover, (3.4) min h |h|∞ λm ≤ Lm1 f ≤ |h|∞ min h λm , for any integer m ≥ 0. (ii) The probability measure ˆ = h ν (this is the so called Gibbs measure generated by f) is σ-inavariant and ˆ = ν ˆ , where ˆ = f − log(h ◦ σ) + log h − log λ . (iii) We have specθ(Lf ) {z ∈ C : |z| = λ} = {λ1, λ2, . . . , λτ } , where λj = λ e2π i j/τ for j = 1, . . . , τ . Moreover each λj is a simple eigenvalue for Lf and every z ∈ specθ(Lf ) with |z| λ satisfies |z|≤ ρ0 λ, where ρ0 can be chosen as follows (3.5) ρ0 = 1 − 1 − θ 4 q2Mτ e 8 τ bθ 1−θ e4M τ |f|∞ 1 Mτ ∈ (0, 1) , and b = bf = max{1, |f|θ}. (iv) For each j = 1, . . . , τ there exists an eigenfunction vj ∈ Fθ(ΣC)+ corresponding to the eigenvalue λj with |vj|θ ≤ 2 h θ and |vj| = h, and a projection operator Pj : Fθ(ΣC) + −→ C · vj, Pj(g) = pj(g) vj, with pj(vj) = 1 and |pj(g)| ≤ |g|∞ min h for any g ∈ Fθ(ΣC), + such that for every g ∈ Fθ(ΣC) + and every integer m ≥ 0 we have (3.6) Lf m g − τ j=1 λj m Pj(g) θ ≤ Df λm ρm g θ , where ρ = √ ρ0 ∈ (0, 1) and (3.7) Df = 108 τ 8 b7 θ10τ (1 − θ)8 q17Mτ e40τ b/(1−θ) e33τM |f|∞ , where b = bf is as in part (iii). Moreover for any g ∈ Fθ(ΣC) + of the form g = τ k=1 Pk(g) we have (3.8) max 1≤k≤τ Pk(g) θ ≤ Hf g θ , where (3.9) Hf = 1 min h 12b q e|f|∞ 1 − θ · h 4 θ (min h)4 τ .

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