Invariant sets and Knaster-Tarski principle

Abstract

Our aim is to point out the applicability of the Knaster-Tarski fixed point principle to the problem of existence of invariant sets in discrete-time (multivalued) semi-dynamical systems, especially iterated function systems. MSC: 54H25, 47H08, 54H20Keywords

Random iteration and projection method

“…Concerning the existence of invariant sets one can assure it under very mild dissipativity conditions even in the absence of continuity of actions, see e.g. the references in [30] or [23]. So let us refine our goal.…”

“…Comparing the nonautonomous discrete dynamical systems ( [27]) with the framework of IFSs (and also multivalued systems, cf. [22,29]), one sees that the orbit of the nonautonomous system is determined by the starting point though the dynamics changes over time and to determine the orbit of the IFS one needs additionally to specify the driving sequence; loosely speaking in the theory of IFSs we deal with the infinite number of nonautonomous systems upon a finite (sometimes countable [32] or compact [39,23,30]) set of generating maps. Yet one can cast the IFS as a skew-product system, e.g., [25,Example 4.3].…”