Note on limit distribution of normalized return times and escape rate

Abstract

In this note we discuss limit distribution of normalized return times for shrinking targets and draw a necessary and sufficient condition using sweep-out sequence in order for the limit distribution to be exponential with parameter $1$. The normalizing coefficients are the same as sizes of the targets. Moreover we study escape rate, namely the exponential decay rate of sweep-out sequence and prove that in $\psi$-mixing systems for a certain class of sets the escape rate is in limit proportional to the size of the set

Entry time statistics to different shrinking sets

“…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.…”

A Poisson limit theorem for Gibbs-Markov maps

“…Consequently, it confirms that the distribution of the first return time tends to an exponential distribution. Note that continued-fraction mixing (or ψ-mixing) is not necessary in this line of proof, whereas a direct calculation of escape rate assuming ψ-mixing but not necessarily spectral gap can be found in [36].…”

Multiple Borel–Cantelli Lemma in dynamics and MultiLog Law for recurrence

“…For more discussion of the distribution of the entry times to small measure sets we refer the readers to [40,97,144,160] and references therein. We also refer to Section 10 for related results in the context of extreme value theory.…”