Abstract
We introduce and study two properties of dynamical systems: topologically transitive and topologically mixing under the set-valued setting. We prove some implications of these two properties for set-valued functions and generalize some results from a single-valued case to a set-valued case. We also show that both properties of set-valued dynamical systems are equivalence for any compact intervals.
Highlights
One of the most important research topics is to determine the chaotic behaviour of the system
Dynamical systems with a topologically transitive property contain at least one point which moves under iteration from one arbitrary neighborhood to any other neighborhood
This property has been studied intensively by mathematicians since it is a global characteristic in the dynamical system
Summary
One of the most important research topics is to determine the chaotic behaviour of the system. Dynamical systems with a topologically transitive property contain at least one point which moves under iteration from one arbitrary neighborhood to any other neighborhood. This property has been studied intensively by mathematicians since it is a global characteristic in the dynamical system. We will introduce and study the notion of topologically transitive and topologically mixing for setvalued functions. We prove that the definitions of these two properties for a set-valued function on compact intervals are equivalence.
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