Limiting Entry and Return Times Distribution for Arbitrary Null Sets

Extreme value distributions of observation recurrences

“…In the case of p = 1 this reverts to the usual Poisson distribution. For more general target sets, the limit law of N n (t) is given by a compound Poisson distribution when the extremal index is different from 1, and by a pure Poisson distribution if no clustering occurs, [37], [38]. We refer also to our paper [15] for a discussion of this matter and related references.…”

“…For a given l, only the terms (l − 1)b 1 and (l − 1)b 2 count in the sum definingα l . By repeating the computation in Lemma 4 in [38] we havẽ…”

“…We are now interested in computing the EI. Since the Cantor set on the y-axis is invariant for the product map T × T on the unit square (the map T was defined in section 4.2), we can adapt the proof "along the diagonal" given by us in [26] section II B or in [22,38], and find easily that only the term q 0 in the expansion of the extremal index will not vanish. Then we use the conformality of the factor measure µ along the x-axis and we get q 0 = 1/2, giving also an extremal index equal to 0.5.…”

Number of visits in arbitrary sets for $ϕ$-mixing dynamics

“…Any compound Poisson distribution depends upon a set of parameters λ l , l ≥ 1. It has been shown in Haydn & Vaienti (2020), that those parameters are related to another sequence α l , l ≥ 1, (see Section 2.2) which quantify the distribution of higher order returns. Whenever the limits defining the α l exist and the latter verify a summable condition, the error term given by the Stein-Chen method will go to zero, and therefore we recover the expected convergence to a compound Poisson law: this is the content of the main result, Theorem 5.…”

“…Applications to concrete examples basically require to check two conditions on the system: (i) first of all the φ, or ψ−mixing property, which enters the estimate of the error in the Stein-Chen approach; (ii) secondly the existence and summability of the α l , which instead depend on the system and on the choice of the nested sequence of small sets U n . A similar program was carried over in Haydn & Vaienti (2020), with a few substantial differences which in particular imply that the examples given in the present paper cannot be covered by the theory developed in Haydn & Vaienti (2020). The latter targets C 2 local diffeomorphisms on smooth manifolds and satisfying a few geometrical and metric conditions, among which the most relevant are: a) local hyperbolicity and distortion; b) the annulus-type condition which allows to control the relative measure of the neighborhoods of the small sets; and finally c) the decay of correlations which is stated in terms of Lipschitz against L ∞ norms.…”