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**Example text**

5 as T (k−1)R γ is good. This gives the proposition. Proof of the Main Theorem, 1. We prove the existence, uniqueness and bounded variation on fibers of the invariant density. 1 applies. Let H : X → R+ be given, some function of bounded variation. Set: hn (ω, x) = (LnT −n ω H )(x) for n ≥ 0. a. t. θ+ . Indeed, the previous proposition gives for all large n, all ≥ 0: θ+ (hn , hn+ ) ≤ ρ n . As hn (ω, ·) 1 = 1, the sequence hn (ω, ·) is also Cauchy in L∞ (m). a. ω. Note that each hn is measurable by (LY0), so that the limit h is measurable.

Definition of a good block. We first give conditions on ω ∈ ensuring that LR ω is a strict contraction of the cone of functions Ca into itself for some appropriate parameter a = a( ) to be defined. 2. 1). 3. For > 0, we define the cone parameter a = a( ) to be: def a = max(6B∗ ( ), 2 · var(1X )). We shall choose R to be a multiple of N and to be so large that e−(λ/2)R < 1/3. 4. t. 4) k=0 essinf LR ω h ≥ α∗ h O( ) stands for functions of such that lim sup Write ∗ for the set of good ω’s. 5) )/ | < ∞.

3, the inequality in (LY1) holds also in this case. As log δ¯ and log K0N are integrable, so is log α N . This in turn implies the integrability of log K N . We compute: log α N dP ≤ log 3 − log δ¯N dP + log δ¯N dP + b 7 ≤ − λ∗ N < 0. 10 This completes the proof of (LY2), hence of (LY). log K0N dP b 34 J. 3. Cones and decay of correlations. Building on the study of the deterministic case by C. Liverani [19], we consider the cone: Ca = {h ∈ L∞ (X, m) : h ≥ 0 (mod m), var(h) ≤ a h 1 } with parameter a to be determined.