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
Lf
m1

|h|∞
min h
λm
,
for any integer m 0.
(ii) The probability measure ˆ ν = h ν (this is the so called Gibbs measure
generated by f) is σ-inavariant and ˆ ν = ν
ˆ
f
, where
ˆ
f = 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
1−θ
e4M
τ |f|∞
1

(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
mg

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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