Some Properties of the Prolongation Limit Random Sets in Random Dynamical Systems

Abstract

 The aim of this paper is to  study the omega limit set with new concepts of  the  prolongation limit random sets in  random dynamical systems, where some  properties are proved and introduced such as the relation among the orbit closure, orbit and omega limit random set. Also we prove that the first prolongation of a closed random set containing this set, the first prolongation is closed and invariant.  In addition, it is connected whenever it is compact provided that the phase space of the random dynamical systems is locally compact. Then, we study the prolongational limit random set  and examined some essential properties of this set. Finally, the relation among the first prolongation, the prolongational limit random set and the positive trajectory of a random set is given and proved.