Existence and Distributional Chaos of Points that are Recurrent but Not Banach Recurrent

Abstract

Recurrence is a classical and basic concept in the study of dynamical systems. According to the 'recurrent frequency' (i.e., the probability of finding the orbit of an initial point entering in a neighborhood), many different levels of recurrent points are found such as periodic points, almost periodic points, weakly almost periodic points, quasi-weakly almost periodic points and Banach recurrent points [19,38,22]. In this paper we are mainly to search the existence on six new levels of recurrent points between Banach recurrence and general recurrence. Moreover, from the perspective of distributional chaos, we will show that five levels of the new recurrent points contain uncountable DC1-scrambled subsets.