A parameter ASIP for the quadratic family

Consider the quadratic family Tₐ(x) = a x (1 - x) for x ∈ [0, 1] and mixing Collet–Eckmann (CE) parameters a ∈ (2,4) . For bounded φ , set φ ₐ := φ - ∫ φ \, dμ ₐ , with μ ₐ the unique acim of Tₐ , and put (σ ₐ (φ ))² := ∫ φ ₐ² \, dμ ₐ + 2 ∑ i>0 ∫ φ ₐ ( φ ₐ ∘ Tⁱₐ) \, dμ ₐ . For any mixing Misiurewicz parameter a* , we find a positive measure set Ω * of mixing CE parameters, containing a* as a Lebesgue density point, such that for any Hölder φ with σ _a*(φ )≠ 0 , there exists ε _φ>0 such that, for normalized Lebesgue measure on Ω *∩ [a*-ε _φ , a*+ε _φ ] , the functions ξ ᵢ(a)= φ ₐ(Tₐⁱ⁺¹(1/2))/σ ₐ (φ ) satisfy an almost sure invariance principle (ASIP) for any error exponent γ>2/5 . (In particular, the Birkhoff sums satisfy this ASIP.) Our argument goes along the lines of Schnellmann’s proof for piecewise expanding maps. We need to introduce a variant of Benedicks–Carleson parameter exclusion and to exploit fractional response and uniform exponential decay of correlations from Baladi et al [Whitney–Hölder continuity of the SRB measure for transversal families of smooth unimodal maps. Invent. Math. 201 (2015), 773–844].