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... often give a substantial amount of information on the statistical properties of the dynamics with respect to µ such as decay of correlations, large deviations, limit theorems, etc.; see for instance =-=[28, 11, 17, 18, 24]-=-. This motivates a general question as to whether any SRB measure µ is of the form (6) for a suitable GMY structure. In this case we say that µ is liftable (to a GMY structure). As a consequence of th...

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... lying in a closed subspace of infinite codimension. Remark 1.2 Since we require β > 1 in (2), our results are restricted to cases where the CLT and related results are known to hold (see for example =-=[20, 21]-=-). An interesting problem is to investigate the case β ∈ (0, 1] which occurs in Example 1.3 below for α ∈ [ 1, 1). Other examples with β = 1 include Bunimovich-type stadia and 2 certain classes of sem...

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...rem, and the law of the iterated logarithm. See [23, 38] and references therein for a survey of consequences of the ASIP. The scalar ASIP has been shown to hold for large classes of dynamical systems =-=[14, 17, 18, 21, 24, 25, 30, 31, 35]-=-. Chernov & Dolgopyat [10, Problem 1] asked for a proof of the ASIP for Rd-valued observables, and it is this problem that is solved in this paper. Our main result applies to a large variety of dynami...

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...ponential tails suffices to deduce many statistical properties of the equilibrium state. These include exponential decay of correlations, the Central Limit Theorem [Y] and Invariance Principles, e.g. =-=[MeN]-=-. Instead of a single potential, thermodynamic formalism makes use of families t ·ϕ of potentials. The occurrence of phase transitions is related to the smoothness of the pressure function t ↦→ P(t · ...

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...Principle, the Local Limit Theorem and the Berry-Esseen Theorem. All these concepts are formally defined in Appendix B, together with the precise form of these Theorems and Principles which we obtain in our setting including values of constants which appear in the statements. The assumption that Corµ(ϕ, ψ ◦ fn) . n−β for some β > 3 implies, by Theorems A and C, that there exists a Gibbs-Markov induced map with m(Rn) . nβ ′ , for some β′ > 2. All the required statistical properties then follow from this by existing results: The Central Limit Theorem [Yo2], the Almost Sure Invariance Principles [MN1, MN2], the Local Limit Theorem and the Berry-Esseen Theorem [Go1]. Interestingly, large deviation properties also follow from the decay of correlations, but they can be proved directly in an abstract measurable setting and do not depend at all on the existence of the induced Gibbs-Markov map. 1.6. Applications. We give a selection of specific systems which satisfy the conditions of some of the Theorems given above. 1.6.1. Large deviations for Viana maps. An important class of nonuniform expanding dynamical systems (with critical sets) in dimension greater than one was introduced by Viana in [Vi]. T...

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...for infinite horizon has recently been established by Szasz and Varju [21] following initial work by Bleher [2]. See also the recent papers by Dolgopyat, Szasz and Varju [11], and Melbourne and Nicol =-=[19]-=-, [20] for related studies of statistical properties of the two-dimensional periodic Lorentz gas. The case of higher dimensions is still open, even for models with finite horizon, cf. Chernov [9], and...

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...(k/n) = Sk/ √ n and interpolating linearly in between. Then Wn(t) weakly converges, as n → ∞, to a Brownian motion (Wiener process) with covariance matrix σ2(A). (c) (Almost sure invariance principle =-=[66, 20]-=-) There exist λ > 0 such that for any A ∈ B̄α we can find (after possibly enlarging the phase space) a Brownian motion (Wiener process) w(t) with variance σ2(A) such that for almost all x there is n0 ...