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 determine the Krieger type of nonsingular Bernoulli actions G g∈G ({0, 1}, µ g ). When G is abelian, we do this for arbitrary marginal measures µ g . We prove in particular that the action is never of type II ∞ if G is abelian and not locally finite, answering Krengel's question for G = Z. When G is locally finite, we prove that type II ∞ does arise. For arbitrary countable groups, we assume that the marginal measures stay away from 0 and 1. When G has only one end, we prove that the Krieger type is always I, II 1 or III 1 . When G has more than one end, we show that other types always arise. Finally, we solve the conjecture of [VW17] by proving that a group G admits a Bernoulli action of type III 1 if and only if G has nontrivial first L 2 -cohomology.

The transfer operator corresponding to a uniformly expanding map enjoys good spectral properties. Here it is verified that coupling yields explicit estimates that depend continuously on the expansion and distortion constants of the map.For nonuniformly expanding maps with a uniformly expanding induced map, we obtain explicit estimates for mixing rates (exponential, stretched exponential, polynomial) that again depend continuously on the constants for the induced map together with data associated to the inducing time.Finally, for nonuniformly hyperbolic transformations, we obtain the corresponding estimates for rates of decay of correlations.

We construct a product measure under which the shift on {0, 1} Z is a type III 1 transformation. IntroductionA conservative non-singular transformation T of the probability space (X, B, P) for which there exists no σ -finite measure which is P-equivalent and T -invariant is said to be of type III [KW, Kri].The first construction of a type III transformation was given in [Orn]. This construction is an odometer and hence does not have the K -property.In [Ham], Hamachi introduced a class of product measures P = ∞ k=−∞ P k under which the full shift is a type III transformation, thus obtaining the first examples of type III transformations that satisfy the K -property.The classification of type III dynamical systems was refined by Krieger [Kri] and Araki-Woods, who introduced the ratio set (see below). Thus type III transformations can be further subdivided into types III λ , with 0 ≤ λ ≤ 1. Krieger showed [KW, Kri] that for each 0 < λ ≤ 1 there is a unique orbit equivalence class.In this work we construct a product measure P = ∞ k=−∞ P k under which the full shift is of type III 1 . This shift is a Markovian extension, in the sense of Silva and Thieullen [ST, Proposition 4.10], of an endomorphism.