Variety of Attractors

In this work, some variation of the concept of attractor are presented, called attractor, physical attractor and proper attractor. As a result, it is proved that the phase space of a topologically exact system is a proper attractor for its inverse and vice versa. Furthermore, in definition of physical attractor, one face with systems in which their phase space attracts all of the open non-empty sets. It is established that a system is topologically mixing if and only if its phase space is physical attractor. Finally, by appealing to IFSs, we demonstrate some classes of non-trivial topologically mixing and topologically exact IFSs.