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... formalism beyond the uniformly hyperbolic context (including countable Markov shift) by several authors (this list is certainly not complete): Araujo [7], Arbieto, Matheus, Oliveira, Varandas, Viana =-=[8, 43, 44, 62, 63]-=-, Bruin, Keller, Todd [17, 16, 18, 19, 20], Buzzi, Paccaut, Sarig, Schmitt [25, 22, 24, 53, 54, 55], Dobbs [34], Denker, Keller, Nitecki, Przytycki, Rivera-Letelier, Urbański [27, 30, 28, 29, 31, 32, ...

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...r that purpose, a measure theoretical weak form of specification is introduced and proved to hold for the robust classes of multidimensional nonuniformly expanding local diffeomorphisms considered in =-=[VV10]-=- and Viana maps [Via97]. 1. Introduction The theory of Large Deviations concerns the study of the rates of convergence at which time averages of a given sequence of random variables converge to the li...

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...tates arise as invariant measures absolutely continuous with respect to an expanding, conformal and non-lacunary Gibbs measure ν. Since we will not use these notions here we shall refer the reader to =-=[VV10]-=- for precise definitions and details. Many important properties arise from the study of transfer operators. We consider the Ruelle-Perron-Fröbenius transfer operator Lf,φ associated to f : M →M and φ...

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... statistics and fluctuations of return times. In fact, one expect the fluctuations of the return times in Theorem A to be log-normal with respect to any measure satisfying a weak Gibbs property as in =-=[26, 23, 24]-=-. Using [19] this is the case provided exponential hitting time statistics. However, to the best of our knowledge, there are no known examples where exponential return time statistics to dynamical bal...

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...s to balls presents itself as a major difficulty in this higher dimension setting. If such a result could be obtained it is most likely that our results can be extended to the more general context of =-=[22]-=-, where [15] is generalized and no Markov partition is assumed to exist. This article is organized as follows. In Section 2 we state our main results. In Section 3 we introduce some notations and tool...

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... examples. The existence and uniqueness of equilibrium states for a broad class of non-uniformly expanding maps in higher dimensions was studied by Oliveira and Viana [OV08] and by Varandas and Viana =-=[VV08]-=-. The multifractal properties of these systems do not appear to have been studied, nor does the question of whether or not these systems (which may have contracting regions) satisfy specification or a...

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...W01]. It can also be applied to non-uniformly hyperbolic systems; in addition to some systems that have already been studied, we describe in Section 7 a class of systems studied by Varandas and Viana =-=[VV08]-=- to which Theorem 2.1 can be applied. Proposition 7.2 gives multifractal results for these systems; these results appear to be completely new. As stated, Theorem 2.1 does not deal with phase transitio...