CHAPTER 3

Preliminaries

Let C be a q × q matrix of 0’s and 1’s and let θ ∈ (0, 1) be a constant. The

matrix C is called irreducible (cf. e.g. Ch. 1 in [PP]) if for all i, j = 1, . . . , q there

exists a positive integer k = k(i, j) such that Ck(i, j) 0, where Ck is a k-fold

product of the matrix C with itself. When C is irreducible, the highest common

divisor τ of all positive integers k such that

Ck(i,

i) 0 for all i = 1, . . . , q is called

the period of C, and the matrix C is called aperiodic if τ = 1. In the latter case

there exists an integer M 0 such that

CM

(i, j) 0 for all i, j.

Denote by C(ΣA)

+

the set of all continuous functions g : ΣA

+

−→ C. Given any

f ∈ C(ΣA),

+

the Ruelle transfer operator Lf : C(ΣA)

+

−→ C(ΣA)

+

is defined by

Lf g(x) =

σ(y)=x

ef (y)

g(y) .

In what follows C will be an irreducible q ×q matrix of 0’s and 1’s. Denote by

τ the period of C. It is known (cf. e.g. [Minc]) that there exists a decomposition

ΣC

+

= X1 ∪ . . . ∪ Xτ of ΣC

+

into a finite disjoint union of closed-open στ -invariant

subsets of ΣC

+

such that for each m = 1, . . . , τ , the map

(στ

)|Xm is isomorphic to

the Bernoulli shift on ΣCm

+

for some aperiodic matrix Cm. Fix a decomposition

with these properties and for each m = 1, . . . , τ let Nm 0 be the minimal positive

integer so that Cmm

N

(i, j) 0 for all i, j = 1, . . . , q. Denote

(3.1) M = max{N1, . . . , Nτ } .

Next, we recall the main parts of Ruelle’s Perron-Frobenius theorem (cf. e.g.

Chapters 2 and 4 in [PP] or Chapter 1 in [Ba]; see also Sect. 1.B in [B] and

Sect. 3 in [AS]). The case of a real valued function f is essentially due to Ruelle

([R1], [R2]), while the complex case essentially follows Pollicott [Po1], [Po2].

The statement of the theorem below is more comprehensive than what is normally

found in the literature and contains some explicit estimates which would be used

below. As one would probably expect, these estimates are relatively rough, and

they are here just to show that the quantities involved can be bound by means of

certain characteristics of the dynamical system σ : ΣC

+

−→ ΣC

+

and the function

f ∈

Fθ(ΣC).+

Throughout we denote by specθ(Lf ) the spectrum of the operator Lf on

Fθ(ΣC).+

Theorem 3.1. (Ruelle-Perron-Frobenius Theorem) Let the matrix C be

irreducible with period τ ∈ IN and let f ∈

Fθ(ΣC).+

(a) Assume that f is real-valued. Then:

(i) There exist a unique λ = λf 0, a probability measure ν = νf on

ΣC+

and a positive continuous function h = hf on ΣC

+

such that Lf h = λ h

and h dν = 1. The spectral radius of Lf : Fθ(ΣC)

+

−→ Fθ(ΣC)

+

is

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