Iterated function systems: transitivity and minimality

Abstract

In this paper, we study the chaotic dynamics of iterated function systems (IFSs) generated by a finite family of maps on a compact metric space. In particular, we restrict ourselves to topological transitivity, fiberwise transitivity, minimality and total minimality of IFSs. First, we pay special attention to the relation between topological transitivity and fiberwise transitivity. Then we generalize the concept of periodic decompositions of continuous maps, introduced by John Banks [1], to iterated function systems. We will focus on the existence of periodic decompositions for topologically transitive IFSs. Finally, we show that each minimal abelian iterated function system generated by a finite family of homeomorphisms on a connected compact metric space X is totally minimal.

Iterated function systems over arbitrary shift spaces

“…Unlike a surjective conventional dynamical system, the last implication in ( 3) is not reversible. This fact was noticed in some literature [16,19]; however, we did not find any example to justify, so we bring our own.…”