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 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

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

.