Poisson Processes for Subsystems of Finite Type in Symbolic Dynamics

Abstract

Let Δ ⊊ V be a proper subset of the vertices V of the defining graph of an irreducible and aperiodic shift of finite type [Formula: see text]. Let ΣΔ be the subshift of allowable paths in the graph of [Formula: see text] which only passes through the vertices of Δ. For a random point x chosen with respect to an equilibrium state μ of a Hölder potential φ on [Formula: see text], let τn be the point process defined as the sum of Dirac point masses at the times k > 0, suitably rescaled, for which the first n-symbols of Tkx belong to Δ. We prove that this point process converges in law to a marked Poisson point process of constant parameter measure. The scale is related to the pressure of the restriction of φ to ΣΔ and the parameters of the limit law are explicitly computed.

Laws of rare events for deterministic and random dynamical systems

“…Next result gives the convergence of the REPP at non-simple points. Note that, contrarily to the usual geometric distribution obtained, for example, in [HV09, CCC09,FFT12a], in here, the multiplicity distribution is quite different. In fact, for eventually aperiodic returning points, for example, we have that π(κ) = 0 for all κ ≥ 3.…”

Entry and return times distribution

“…[15]). This approach was more conclusively carried out by Coelho and Collet [24] and also in [20] for measures on subshifts that have strong mixing properties. The combinatorics involved tend to make such an approach difficult and would favour the moment method.…”

Entry time statistics to different shrinking sets

“…We prove that for µ a Gibbs measure of Hölder potential, and under suitable conditions, that depend on µ and the topological entropy of (X , σ) and (Y, σ), the sequence X n converges in law to an exponential random variable. This is closely related to the results in [9] but our proofs are motivated by [11]. Recently, a necessary and sufficient condition for X n to converge in law to an exponential random variable when µ is an ergodic probability measure is established in Theorem 1 in [25], however, to check this condition requires not straightforward estimations.…”

“…In this paper we consider the problem of finding conditions on the pair ({U n }, µ) so that X n converges in law to an exponential random variable when µ(U n ) → 0 as n → ∞, but the sequence {U n } does not necessarily shrink to a single point (or finitely many points). The conditions we impose on U n and µ are similar to the used in [9,18,19]. In many cases these conditions are easy to verify, this allows us to find new systems satisfying exponential limit distribution.…”