Answering two open problems on Banks theorem for non-autonomous dynamical systems

TL;DR

This paper addresses two open problems in non-autonomous dynamical systems by constructing examples that challenge existing assumptions about transitivity, periodic orbits, and sensitivity, thus advancing the theoretical understanding of these systems.

Contribution

It provides explicit examples of non-autonomous dynamical systems that exhibit novel properties, answering open questions about their structure and behavior.

Findings

Existence of a topologically transitive NADS with two disjoint invariant periodic orbits, finitely generated but not periodic.

Existence of a topologically transitive, non-finitely generated NADS with two disjoint invariant periodic orbits, not sensitive.

Answers to Open Problems 4.1 and 4.2 in the literature.

Abstract

This paper shows that (1) there exists a topologically transitive NADS having two disjoint invariant periodic orbits with dense periodic points, which is finitely generated but not periodic; (2) there exists a topologically transitive non-finitely generated NADS having two disjoint invariant periodic orbits with dense periodic point, which is not sensitive. This answers positively the Open Problems 4.1 and 4.2 posed in [8].