On Equicontinuity and Related Notions in Nonautonomous Dynamical Systems

TL;DR

This paper studies properties like equicontinuity, minimality, and transitivity in non-autonomous dynamical systems generated by commutative families of homeomorphisms, correcting previous results and extending known autonomous system results.

Contribution

It corrects a false claim about minimality conditions, establishes new results on almost periodic points, and characterizes transitivity and minimality in non-autonomous systems.

Findings

Corrects previous minimality conditions for non-autonomous systems.

Shows that almost periodic points imply uniform almost periodicity in equicontinuous flows.

Characterizes transitivity and minimality in terms of dense orbits and sensitivity.

Abstract

In this work, we investigate the dynamics of a general non-autonomous system generated by a commutative family of homeomorphisms. In particular, we investigate properties such as periodicity, equicontinuity, minimality and transitivity for a general non-autonomous dynamical system. In \cite{sk2}, the authors derive necessary and sufficient conditions for a system to be minimal. We claim the result to be false and provide an example in support of our claim. Further, we correct the result to derive necessary and sufficient conditions for a non-autonomous system to be minimal. We prove that for an equicontinuous flow generated by a commutative family, while the system need not exhibit almost periodic points, if $x$ is almost periodic then every point in $\overline{\mathcal{O}_H(x)}$ is almost periodic. We further prove that in such a case, the set $\overline{\mathcal{O}_H(x)}$ is uniformly almost periodic and hence provide an analogous extension to a result known for the autonomous systems. We prove that a system generated by a commutative family is transitive if and only if it exhibits a point with dense orbit. We also prove that any minimal system generated by commutative family is either equicontinuous or has a dense set of sensitive points.