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By A. Jaffe (Chief Editor)

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V is finite and cases (1) and (2) do not hold, 4. V = SL(2, C). In the last case the identity component V ◦ of V coincides with the whole V . For a general second order linear differential equation y¨ = ry, with r a meromorphic function over P1 , necessary conditions for the above cases to hold are recovered in the following proposition (see the first theorem in Sect. 1 of [17]). 2. 1 to hold are 1. Every pole of r must have even order or else have order 1, and the order of r at ∞ must be even or else be greater than 2 in order for the case (1) to hold 2.

All n ≥ n0 (ω), all ≥ 0 and all |p| ≤ n: θ+ (LnT p ω h, Ln+ g) ≤ ρ n T p− ω for all h, g : X → R+ with m(h) = m(g) = 1 and var(h) ≤ V (ω)e n , var(g) ≤ V (ω)e (n+ ) . Moreover ρ = ρ( ) does not depend on V . Proof of the Proposition. 1 so that: > 0 be given. a. ω ∈ . > 0 defines B∗ , R, a (see Sect. 2). Let n, ≥ 0 and |p| ≤ n. Set j∗ = j∗ (T p ω) (see Sect. 1). Let d be the smallest integer ≥ 0 satisfying: 1 (1) j∗ + dR ≥ λ− ( n + log V (ω)); (2) C0 (T p+j∗ +dR ω) ≤ B∗ . = O( )n/R integers d in the These two conditions fail for at most O( ) |p|+|p+n| R interval [−|p|/R, |p + n|/R] using the definition of j∗ and assuming that n is large.

To get back to ϕ, remark that: µγ (ϕ)(var(φ) + 1) = (var(ϕ) + µγ (ϕ)) ≤ (var(ϕ) + ϕ ≤ (Cvar + 1)(var(ϕ) + ϕ 1 ). ∞) We may thus conclude that for large n: |Cγ (ϕ, ψ, n)| ≤ C(ω)(var(ϕ) + ϕ 1 ) ψ ∞ (ρe s n ) , and recall that: ρes < 1. Finally this estimate extends to all n ≥ 0 by enlarging C(ω): it is enough to recall that |Cγ (ϕ, ψ, n)| ≤ 2 ϕ ∞ ψ ∞ ≤ 2Cvar · (var(ϕ) + ϕ 1 ) ψ ∞ . Appendix A. Counter-Examples Example 1: Slow decay of integrated correlations. s. interval map f satisfying (A0)–(A4) such that if ϕ(x) = 2 · 1[0,1/2] (x), then the integrated correlation function satisfies: Cω (ϕ, ϕ, n) dP ∼ const · n−δ as n → ∞ def Cint (ϕ, ϕ, n) = (a ∼ b meaning that lim a/b = 1).

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


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