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**Extra resources for Communications in Mathematical Physics - Volume 208**

**Example text**

We have, for n < nc (ω, i): C(ω,i) (ϕ, ϕ, n) = 1 ≤ K(ω, i) · 2 · 2 · ρ n . Thus, log K(ω, i) ≥ | log ρ|nc (ω, i) + const. But: nc dP = P(nc > n) = n≥0 C(ω,i) (ϕ, ϕ, n) dP = n≥0 Cint (ϕ, ϕ, n) = ∞, n≥0 because of Example 1. Example 3: Mixing without covering. s. m. a. t. the skew-product F , (3) f does not satisfy the covering assumption (A4). More precisely, (3’) there exists ϕ : [0, 1] → R with bounded variation such that: Cω (ϕ, ϕ, n) = ±1 for all n ≥ 0. , pathwise) mixing. Construction. Let ( , A, P, T ) be the ( 21 , 21 )-Bernoulli shift on {−1, +1}Z .

There exists nc = nc (ω, B) < ∞ such that, for all n ≥ nc , f n (B) = X modulo Lebesgue measure. Then f defines a good random transfer operator. 2. Let f1 , f2 , . . , fn be multi-dimensional β-transformations. Write (X, P (n) , f (n) ) for the piecewise affine map defined by f (n) = fn ◦ · · · ◦ f1 . Then: mult(∂P (n) ) ≤ (n + 1) · d. Proof. This is a corollary of the proof of Lemma 1 in [8]. 3 ([8, Lemma 5]). Let f : Y → X be a β-transformation. If B ⊂ [0, 1[d is a ball with radius r then f (B) either is the whole [0, 1[d or it contains a ball of radius δ(f√) r.

Therefore h = g (mod m). This proves the uniqueness of the globally invariant density. 2. a. ω ∈ . a. ω and all q ∈ Z: hT q ω ∞ + var(hT q ω ) ≤ C(ω)es|q| . log K dP, K given by (LY1). Without loss Proof of the lemma. Let s > 0. Set σ0 = of generality, we assume that s/2(σ0 + 1) < 1. , n ≥ n0 (s)) and each choice of the sign ±: log K(T ±j w) ≤ (σ0 + 1)n. a. w ∈ , for A = A(s) < ∞ large enough, P({w : var(hw ) > A}) < 4(σ0s+1) . a. ω and each choice of ±: lim k→±∞ 1 #{0 ≤ j < |k| : var(hT ±j ω ) > A or T ±j ω ∈ / |k| 2} < s < 1.