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(Σ+) A the set of all continuous functions g : Σ+ A −→ 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 Σ+ C = X1 . . . of Σ+ C into a finite disjoint union of closed-open στ -invariant subsets of Σ+ C such that for each m = 1, . . . , τ , the map (στ )|X m is isomorphic to the Bernoulli shift on Σ+ Cm 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(i, N 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 : Σ+ C −→ Σ+ C 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 Σ+ C and a positive continuous function h = hf on Σ+ C such that Lf h = λ h and h = 1. The spectral radius of Lf : Fθ(ΣC) + −→ Fθ(ΣC) + is 9
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