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

TL;DR

This paper explores the nuanced hierarchy of recurrent points in dynamical systems, revealing complex topological structures and distributional chaos among points that are recurrent but not Banach recurrent, especially in symbolic dynamics.

Contribution

It introduces six refined recurrence levels between Banach recurrence and general recurrence, demonstrating their topological complexity and distributional chaos in symbolic and expansive systems.

Findings

Six refined recurrence levels exist with strong topological complexity.

All refined levels are dense and contain uncountable distributionally chaotic sets.

Points recurrent but not Banach recurrent exhibit distributional chaos in various systems.

Abstract

According to the recurrent frequency, many 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. In this paper, we consider symbolic dynamics and show the existence of six refined levels between Banach recurrence and general recurrence. Despite the fact that these refined levels are all null-measure under any invariant measure, we further show they carry strong topological complexity. Each refined level of recurrent points is dense in the whole space and contains an uncountable distributionally chaotic subset. For a wide range of dynamical systems, such as expansive systems with the shadowing property, we also show the distributional chaos of the points that are recurrent but not Banach recurrent.