A note on multi-transitivity in non-autonomous discrete systems

TL;DR

This paper investigates stronger forms of transitivity in non-autonomous discrete systems, providing counterexamples to previous theorems, introducing mildly mixing, and establishing new relationships among transitivity, mixing, and chaos.

Contribution

It corrects prior inaccuracies, introduces the concept of mildly mixing, and clarifies the implications of multi-transitivity in non-autonomous systems.

Findings

Multi-transitivity implies Li-Yorke chaos.

Mildly mixing implies multi-transitivity.

Counterexamples show previous theorems are incorrect.

Abstract

This paper is concerned with some stronger forms of transitivity in non-autonomous discrete systems$(f_{ 1,\infty})$ generated by a uniformly convergent sequence of continuous self maps. Firstly, we present two counterexamples to show that Theorem 3.1 obtained by Salman and Das in [Multi-transitivity in nonautonomous discrete systems Topol. Appl. 278(2020)107237] is not true. Then, we introduce and study mildly mixing in non-autonomous discrete systems, which is stronger than mixing. We obtain that multi-transitivity implies Li-Yorke chaos and that mildly mixing implies multi-transitivity, which answer the open problems 1 and 2 in the paper above. Additionally, we give a counterexample which shows that Theorem 2.3 and Theorem 2.4 given by Sharma and Raghav in [On dynamics generated by a uniformly convergent sequence of maps Topol. Appl. 247 (2018)81-90] are both incorrect and give the correct proofs of them. Finally, some counterexamples are constructed justifying that some results related to stronger forms of transitivity which are true for autonomous systems but fail in non-autonomous systems, and establish a sufficient condition under which the results still hold in non-autonomous systems.