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Download Communications in Mathematical Physics - Volume 208 by A. Jaffe (Chief Editor) PDF

By A. Jaffe (Chief Editor)

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Claim. f satisfies (LY). Proof. (LY0) has been checked already. 3 and (A0)-(A3) if we define K(ω) = K0 (ω) as in the statement of the lemma (in particular, log+ K0 is integrable). It remains to check (LY2) as (LY3) will be a consequence of the rest. By (A1), assuming that K∗ < ∞ is a large constant, we have: def λ∗ = log min(δ, K∗ ) dP > 0. Choose an integer N > 0 so large that log 3 < λ∗ N/10. def def N k Set δ¯N (ω) = N−1 k=0 min(δ(T ω), K∗ ) and K0 (ω) = so small that for any E ⊂ with P(E) < 0 : E log K0N dP < λ∗ N and 10 E N −1 k k=0 K0 (T ω).

5 below. 1. Construction of the coating intervals. 2. For ω ∈ choice of the sign ±: (i) limm→∞ (ii) limm→∞ 1 m #{0 1 m #{0 , j∗ (ω) is the smallest integer 0 ≤ j∗ < R such that, for each ≤ k < m : T ±kR+j∗ ω ∈ ∗ } > 1 − , ≤ k < m : C0 (T ±kR+j∗ ω) ≤ B∗ } > 1 − . e. as at least (3/4)R integers j∗ in [0, R[ satisfy each one of these conditions. Obviously j∗ is a measurable function with: j∗ (T j∗ (ω) ω) = 0 and j∗ (T R ω) = j∗ (ω). Now, for ω ∈ n ≥ 1 such that: , define the coating length (ω) to be, if ω is bad, the smallest integer 1 n log K R (T kR ω) ≤ 1/2 σR 0≤k

Z/n2+δ , . . ) (Z being the normalization constant). Let h(ω) = ω0 (the 0th coordinate of ω ∈ ˜ ). ˜ h d P˜ = n≥1 Z/n1+δ < ∞. : ˜ P, Let ( , A, P, T ) be the suspension by h over ( ˜ , A, = {(ω, n) ∈ ˜ × N : 0 ≤ n < h(ω)}, T (ω, n) = (ω, n + 1) if n + 1 < h(ω) and T (ω, h(ω) − 1) = (T˜ ω, 0), P(A) = hd P˜ −1 ˜ ∩ ( ˜ × {n})). P(A n≥0 Decay of Correlations for Random Lasota–Yorke Maps 47 Write E(·) for the integer part and {·} for the fractional part. : g0 (x) = {2x}. : g1 (x) = 1 2 (E(2x) + {4x}).

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