On the Complexity of Expansive Measures of Nonautonomous Dynamical Systems

In this paper, for one thing, we introduce positively expansive system and positively expansive measure for nonautonomous dynamical systems. And we study the relation between positively expansive measure and the maximal cardinalities of separated set. We obtain the following main results. Firstly, we prove that every nonautonomous dynamical system with positively expansive measure implies there are e > 0 and a sequence of (n, e)-separated sets whose cardinalities go to infinite as n -> infinity. Secondly, we point out that every positively expansive system also implies the above properties. Thirdly, for each nonautonomous dynamical system that satisfies the above properties, there is a Borel probability measure such that the measure of the set of stable points of nonautonomous dynamical systems is zero. For another, we introduce positively countably expansive and positively measure expansive for nonautonomous dynamical systems. In addition, we prove that a nonautonomous dynamical system is positively measure expansive if and only if it is positively countably expansive. Finally, we introduce stable points and equicontinuous for a nonautonomous dynamical system and study some relevant properties. This paper primarily generalizes the main results obtained by Morales to nonautonomous dynamical systems.