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...∈[a,b] Kf,ψ(s) (6.2) and lim inf n→∞ 1 n log νf,φ ( x ∈M : 1 n Snψ(x) ∈ (a, b) ) ≥ − inf s∈(a,b) Kf,ψ(s) (6.3) where Kf,ψ(s) = − sup {−Ptop(f, φ) + hη(f) + ∫ φdη : ∫ ψ dη = s}. We refer the reader to =-=[Va12]-=- for a proof of the previous assertions and extension for weak Gibbs measures. Moreover, if ψ is Hölder continuous then Theorem F yields a large deviation principle where the rate function Kf,φ in (6...

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...roduced by Pfister & Sullivan [29] and Thompson [33] to the study of multifractal formalism for Birkhoff averages associated to beta-shifts, and by Pfister & Sullivan [28], Yamamoto [37] and Varandas =-=[34]-=- to study large deviations. This article will use the weak form of specification introduced by Varandas [34] in a nonuniformly hyperbolic context. Denote by M(X) and M(X, T ) the set of all Borel prob...

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...level-2 results via the contraction principle (see [10]). To place our results in context, we summarize the main approaches to large deviations in dynamical systems. We recommend the introductions of =-=[41, 27, 35, 10, 5]-=- for further references and discussion. The Large Deviation Principle can be divided into two steps: (1) Upper Bound: bound lim 1 n logm{x | En(x) ∈ U} from above in terms of a rate function. For an o...

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...s do not satisfy both the specification and shadowing properties. On the other hand, several authors considered more recently either measure theoretical non-uniform specification properties (see e.g. =-=[16, 23]-=-) or almost specification properties (see e.g. [17, 22]) to the study of the ergodic properties of a given dynamical system. One remaining interesting question is to understand which partially hyperbo...

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...roduced by Pfister & Sullivan [21] and Thompson [27] to the study of multifractal formalism for Birkhoff averages associated to beta-shifts, and by Pfister & Sullivan [20], Yamamoto [31] and Varandas =-=[28]-=- to study large deviations. This article will use the weak form of specification introduced by Varandas [28] in a nonuniformly hyperbolic context. Denote by M(X) and M(X, T ) the set of all Borel prob...

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...resurgence in interest in specification properties recently, due to important contributions by Pfister and Sullivan (almost specification [PS07, Yam09, Tho10]), and Varandas (nonuniform specification =-=[Var10]-=-). Our work is very much in the spirit of these developments, giving another direction in which to weaken the specification property in order to apply to a wider range of examples. In §2, we collect o...