Exponential return times in a zero-entropy process

,Paulina Grzegorek,Michal Kupsa

doi:10.3934/cpaa.2012.11.1339

Abstract

We construct a zero-entropy weakly mixing finite-valued process with the exponential limit law for return resp. hitting times. This limit law is obtained in almost every point, taking the limit along the full sequence of cylinders around the point. Till now, the exponential limit law for return resp. hitting times, taking the limit along the full sequence of cylinders, have been obtained only in positive-entropy processes satisfying some strong mixing conditions of Rosenblatt type.

Highlights

In the last two decades, asymptotic laws for the return and hitting time statistics in stationary processes were intensively studied
In a general situation, this ensures that both distributions are close to the exponential one
We find this approach very useful, especially in the case where the process exhibits some kind of mixing behavior

Summary

Introduction

In the last two decades, asymptotic laws for the return and hitting time statistics in stationary processes were intensively studied. There are several classes of zero-entropy processes, which do not satisfy these strong mixing conditions and possess another limit distribution for return Chaumoitre and Kupsa [5] proved that in the class of processes derived from rank-one systems, one can obtain any possible limit law for return and hitting times satisfying some very weak and natural condition described in [12] and [11].

PAULINA GRZEGOREK AND MICHAL KUPSA

Take k

We simplify the notation and denote x n

We will prove that

Since u