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 . . . 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, . . . , } .
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 = 1. The spectral radius of Lf : Fθ(ΣC)
+
−→ Fθ(ΣC)
+
is
9
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