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...We point out that in the case of interval maps with nonzero Lyapunov exponent, Saussol, Troubetzkoy and Vaienti prove that R = HD(µ) µ-a.e. for ergodic measures, under very weak regularity conditions =-=[14]-=-. See Remark 17-(i) for related results. We now give a sketch of the strategy adopted in this paper. Theorem 8 states that under sufficiently rapid mixing the recurrence rates equal the pointwise dime...

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...e will assume it in this paper. It is interesting that "return times to balls" recently allowed to reformulate several results of thermodynamic formalism in terms of return times statistics =-=only, see [16, 2, 32, 13]. In [32]-=- this approach was called "thermodynamics of return times". Assume that X is a Riemannian manifold, consider an open set ^ X X and let ^ f : ^ X ! ^ X thesrst return map. We will denote by ...

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...e note that (2) was proved to hold for dynamical systems that either present some hyperbolicity (e.g. Axiom A diffeomorphisms [4] and piecewise monotone interval maps whose derivative has p-variation =-=[21]-=-) or that satisfy a rapidly mixing property as in [20]. In [21, 22] the minimal recurrence rates S(x) = limsup r→0 S(B(x, r)) − log r and S(x) = liminf r→0 S(B(x, r)) − logr are studied. In the case o...

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...he Hausdorff measure of X and other authors have also obtained new quantitative recurrence results relating various recurrence indicators with entropy and dimension, see e.g. [2], [6], [24], [25] and =-=[41]-=-. It is natural to ask if the orbit {T n (x)} of the point x comes back not only to every neighborhood of x itself as Poincaré’s Theorem asserts, but whether it also visits every neighborhood of a pre...

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...〈τ(U)〉 versus (− ln ɛ) and evaluating the slope b q0 . q0 Furthermore, the relation in (3.4) implies that the dynamical system (Rd,f) possesses positive topological entropy [3]. On the other hand, in =-=[21]-=-, it was proved that the Lyapunov exponent of some class of f can be estimated from the behavior of the first return times of a ball as the diameter vanishes. More precisely, if f is a piecewise monot...

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...quickly accessible in experiments. In addition, relevant quantifiers of low-dimensional chaotic systems can be obtained by the statistical properties of the FPR. The dimensions and Lyapunov exponents =-=[3, 4, 5, 6, 7]-=-, the multifractal spectrum [5], the correlation function [8, 9] and the extreme value laws [10]. However, for most of the rigorous results concerning the FPR, in particular the form of the DFP [11, 1...

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...y the Birkhoff ergodic theorem there exist constants r0, r1 ∈ [0,∞] such that for almost all x ∈ X , Rξ(x) = r0 and Rξ(x) = r1. The ergodic case has been treated in several more papers. Saussol et al =-=[9]-=-( see also [1]) show that if the entropy of µ is positive, then r0 ≥ 1. Cassaigne et al [2] show that this inequality is not satisfied for systems with zero entropy. In particular for the Fibonacci sh...

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...ta analysis were described. In addition, relevant quantifiers of low-dimensional chaotic systems may be obtained by the statistical properties of the FPR such as the dimensions and Lyapunov exponents =-=[7, 8]-=- and the extreme value laws [9]. For most of the rigorous results concerning the FPR, in particular the form of the DFP [10], one needs to consider very long returns to arbitrarily small regions in ph...