We consider a family of Pomeau-Manneville type interval maps T α , parametrized by α ∈ (0, 1), with the unique absolutely continuous invariant probability measures ν α , and rate of correlations decay n 1−1/α . We show that despite the absence of a spectral gap for all α ∈ (0, 1) and despite nonsummable correlations for α ≥ 1/2, the map α → ϕ dν α is continuously differentiable for ϕ ∈ L q [0, 1] for q sufficiently large.

A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP url' above for details on accessing the published version and note that access may require a subscription. AbstractWe prove statistical limit laws for sequences of Birkhoff sums of the typewhere T n is a family of nonuniformly hyperbolic transformations. The key ingredient is a new martingale-coboundary decomposition for nonuniformly hyperbolic transformations which is useful already in the case when the family T n is replaced by a fixed transformation T , and which is particularly effective in the case when T n varies with n.In addition to uniformly expanding/hyperbolic dynamical systems, our results include cases where the family T n consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters), Viana maps, and externally forced dispersing billiards.As an application, we prove a homogenization result for discrete fast-slow systems where the fast dynamics is generated by a family of nonuniformly hyperbolic transformations.

We prove the one-dimensional almost sure invariance principle with essentially optimal rates for slowly (polynomially) mixing deterministic dynamical systems, such as Pomeau-Manneville intermittent maps, with Hölder continuous observables.Our rates have form o(n γ L(n)), where L(n) is a slowly varying function and γ is determined by the speed of mixing. We strongly improve previous results where the best available rates did not exceed O(n 1/4 ).To break the O(n 1/4 ) barrier, we represent the dynamics as a Young-towerlike Markov chain and adapt the methods of Berkes-Liu-Wu and Cuny-Dedecker-Merlevède on the Komlós-Major-Tusnády approximation for dependent processes. Theorem 1.3. Let γ ∈ (0, 1/2) and ϕ : [0, 1] → R be a Hölder continuous observable with ϕ dµ = 0. For the map (1.1), the random process S n (ϕ) satisfies the ASIP with variance c 2 given by (1.4) and rate o(n γ (log n) γ+ε ) for all ε > 0. For the map (1.5), the random process S n (ϕ) satisfies the ASIP with variance c 2 given by (1.4) and rate o(n γ ).The rates in Theorem 1.3 are optimal in the following sense:Proposition 1.4. Let f be the map (1.1). There exists a Hölder continuous observable ϕ with ϕ dµ = 0 such that lim sup n→∞ (n log n) −γ |S n (ϕ) − W n | > 0 for all Brownian motions (W t ) t≥0 defined on the same (possibly enlarged) probability space as (S n (ϕ)) n≥0 . Hence, one cannot take ε = 0 in Theorem 1.3.Remark 1.5. If c 2 = 0, the rate in the ASIP can be improved to O(1). Indeed, then it is well-known that ϕ is a coboundary in the sense that ϕ = u−u•f with some u : [0, 1] → R. By [7, Prop. 1.4.2], u is bounded, thus S n (ϕ) is bounded uniformly in n.Remark 1.6. It is possible to relax the assumption that ϕ is Hölder continuous. As a simple example, Theorem 1.3 holds if ϕ is Hölder on (0, 1/2) and on (1/2, 1), with a discontinuity at 1/2. See Subsection 4.3 for further extensions.Remark 1.7. Intermittent maps are prototypical examples of nonuniformly expanding dynamical systems, to which our results apply in a general setup, and so does the discussion of rates preceding Theorem 1.3. We focus on the maps (1.1) and (1.5) for simplicity only, and discuss the generalization in Section 5.