# Invariance of distributional chaos for backward shifts

## TL;DR

This paper establishes a precise condition under which the backward shift operator on K"{o}the sequence spaces admits an invariant distributionally $ ext{epsilon}$-scrambled set, advancing previous results in the area.

## Contribution

It provides a necessary and sufficient condition for the existence of invariant distributionally $ ext{epsilon}$-scrambled sets for backward shifts on K"{o}the spaces, improving earlier findings.

## Key findings

- Characterization of invariant distributionally $ ext{epsilon}$-scrambled sets
- Improved conditions compared to previous work
- Enhanced understanding of chaos invariance in sequence spaces

## Abstract

A sufficient and necessary condition ensuring that the backward shift operator on the K\"{o}the sequence space admits an invariant distributionally $\varepsilon$-scrambled set for some $\varepsilon>0$ is obtained, improving the main results in [F. Mart\'{\i}nez-Gim\'{e}nez, P. Oprocha, A. Peris, J. Math. Anal. Appl., {\bf 351} (2009), 607--615].