Metric and Topological Multivalued Fractals

Abstract

Applying metric (Banach-like) and topological (Schauder-like) fixed-point theorems, the existence of metric and topological fractals is respectively proved as (sub)invariant subsets of the Hutchinson–Barnsley map generated by a multifunction system. Weakly contractive and compact multifunction systems are considered, but systems of more general multifunctions are discussed as well. The notions of hyperspaces and AR-spaces are employed for this goal.

On the Continuity of the Hutchinson Operator

“…However, simple examples show that an upper semicontinuous multifunction on a compact space need not induce a continuous Hutchinson operator, e.g., [4], (Counter-Example 1) and [5], (Proposition 1.5.3).…”

“…It is well known that the Hutchinson operator inherits essentially all of the continuity properties of the functions of the underlying iterated function system (IFS), cf. [4][5][6]. However, some issues remain obscure and unexplored.…”

A mathematical–physical approach on regularity in hit-and-miss hypertopologies for fuzzy set multifunctions

“…We also mention different aspects concerning generalized fractals in hyperspaces endowed with Hausdorff, or, more generally, with Vietoris hypertopology (see Andres and Fi šer [2], Andres and Rypka [3], Banakh and Novosad [5], Kunze et al [21]). …”

Invariant sets and Knaster-Tarski principle

“…[4,47,60,64]. The main drawback of the fixed point theorems on ordered sets compared to the Banach-Caccioppoli contraction principle is the lack of uniqueness of a fixed point.…”

“…Existence of an invariant set is established there via two methods: symbolic dynamics and contraction principle. Both approaches became a basis for further extensions: [47,59] (symbolic dynamics), [4,34,64,76,77] (contraction principle). Moreover, Hutchinson related invariant sets of IFSs to supports of invariant measures which proved to be a fruitful line of investigations: [12,27,51,59,70].…”