# Reiterative distributional chaos on Banach spaces

## TL;DR

This paper introduces new notions of reiterative distributional chaos for linear operators on Banach spaces, linking these concepts to irregular vectors and exploring their relationships with other dynamical properties.

## Contribution

It defines and characterizes reiterative distributional chaos types for Banach space operators, expanding the understanding of chaos in linear dynamics.

## Key findings

- Reiterative distributional chaos types are characterized by irregular vectors.
- Conditions for the existence of subspaces with vectors that are irregular and distributionally near zero.
- Connections established between reiterative chaos and other dynamical properties.

## Abstract

If we change the upper and lower density in the definition of distributional chaos of a continuous linear operator on Banach space by the Banach upper and Banach lower density, respectively, we obtain Li-Yorke chaos. Motivated by this fact, we introduce the notions of reiterative distributional chaos of types $1$, $1^+$ and $2$ for continuous linear operators on Banach spaces, which are characterized in terms of the existence of an irregular vector with additional properties. Moreover, we study its relations with other dynamical properties and give conditions for the existence of a vector subspace $Y$ of $X$ such that every non-zero vector in $Y$ is both irregular for $T$ and distributionally near to zero for $T.$