Dynamics of weakly mixing non-autonomous systems

TL;DR

This paper investigates the relationship between topological transitivity, weak mixing, and chaos in non-autonomous dynamical systems, establishing equivalences and providing counterexamples to autonomous system analogs.

Contribution

It establishes the equivalence of transitivity and weak mixing in non-autonomous systems and explores chaos and sensitivity conditions, extending classical autonomous results.

Findings

Topological transitivity on probability measures is equivalent to weak mixing in non-autonomous systems.

Counterexamples show some autonomous results do not extend to non-autonomous systems.

Weakly mixing systems on intervals imply Devaney chaos under periodic conditions.

Abstract

For a commutative non-autonomous dynamical system we show that topological transitivity of the non-autonomous system induced on probability measures (hyperspaces) is equivalent to the weak mixing of the induced systems. Several counter examples are given for the results which are true in autonomous but need not be true in non-autonomous systems. Wherever possible sufficient conditions are obtained for the results to hold true. For a commutative periodic non-autonomous system on intervals, it is proved that weakly mixing implies Devaney chaos. Given a periodic non-autonomous system, it is shown that sensitivity is equivalent to some stronger forms of sensitivity on a closed unit interval.