On Sevast’yanov's theorem

Poisson Processes for Subsystems of Finite Type in Symbolic Dynamics

“…In fact, given any aperiodic ergodic dynamical system, one can get any limit law within a large class of laws [18] by using a suitable family of shrinking neighbourhoods. In contrast with this abstract result, when one considers cylinder sets about a generic point of a system mixing "sufficiently well" its partition, one expects and gets an exponential limit law for the first hitting time, and a Poisson law for the hitting time process; see for instance the papers [1,6,7,10,11,12,13,14,15,16,17,22]. Here, as in [5], we consider another case where the intersection of the shrinking neighbourhoods contains a non-trivial invariant set and we show that a marked Poisson point process appears as the asymptotic limit law.…”

“…Let ∆ n be defined by (11) and consider the hitting time point process τ n of ∆ n rescaled by c −1 n of (1), where c n is to be suitably chosen below. The aim is to show that Hypotheses (H.1-2) are satisfied and then to conclude convergence in law of τ n to a marked Poisson point process, identifying the parameters λ and π j .…”

“…This proves Proposition 4 as stated in the present paper. For the remaining comments in Section 2, we mention the corresponding changes in the expressions ( 9), (10) and (11) of the main result of [8] A . First we note that for f, g ∈ L 1 (µ) we have for every n 1, L n…”

“…Putting g = χ B , f = h ∆ in the above expression and noting that L n ∆ (h ∆ ) = e nP ∆ h ∆ we obtainµ P Y (T −n B ∩ ∆ n ) = T −n B∩∆n h ∆ dµ = ∆n h ∆ • χ B •T n dµ = χ B • L n ∆ (h ∆ ) dµ = e nP ∆ χ B • h ∆ dµ = e nP ∆ µ P Y (B) .This proves(9), since for n = 1 we obtainµ P Y (T −1 B ∩ ∆) = e P ∆ µ P Y (B) ,where we identified ∆ with the set ∆ 1 . Putting g = χ B and f = 1 in(22) givesµ(T −n B ∩ ∆ n ) µ(∆ n ) = ∆n χ B •T n dµ ∆n dµ = χ B • L n ∆ (1) dµ L n ∆ (1) dµ = B e −nP ∆ L n ∆ (1) dµ e −nP ∆ L n ∆ (1) dµ .Taking the limit when n → ∞ proves(11), sincelim n→∞ µ(T −n B | ∆ n ) = B h ∆ dµ h ∆ dµ = µ P Y (B) µ P Y (Σ + A ).Although Σ ∆ (which is the support of µ ∆ ) has µ-measure zero, an interesting fact is that(23) µ ∆ (B) = lim n→∞ µ(B | ∆ n ) ,for every closed and open subset B ⊆ Σ + A . (Since µ and µ ∆ are ergodic measures for T , they are mutually singular, therefore the above is untrue in general for all Borel sets B.)…”

“…We note that ∆ n ⊆ ∆ n−1 for each n > 0 and ∩ n>0 ∆ n = Σ ∆ . The Pianigiani-Yorke measure also satisfies (11) µ P Y (B)…”