Contractive function systems, their attractors and metrization

Abstract

Abstract. In this paper we study the Hutchinson-Barnsley theory of fractals in the setting of multimetric spaces (which are sets endowed with point separating families of pseudometrics) and in the setting of topological spaces. We find natural connections between these two approaches.

Topological version of generalized (infinite) iterated function systems

“…0 and a 2 ðs aj k ðP k ÞÞ for every k 2 N, we have that T k2N s aj k ðP k Þ ¼ fag, and hence T k2N f aj k ðXÞ ¼ fgðaÞg, so it is a singleton and ðt 2 Þ holds, and also g ¼ p. In particular, pðXÞ ¼ gðXÞ ¼ X, so ðt 3 Þ holds trivially. h In the case of finite (i.e., consisting of finitely many mappings) TIFSs (see Section 2.8), the following remetrization result holds ( [1,21]):…”

“…It was proved in [16,1] that every IFS consisting of Matkowski contractions is a TIFS and Theorem 2.20. Each TIFS generates a unique proper fractal.…”

On a typical compact set as the attractor of generalized iterated function systems of infinite order

“…(called sometimes a fibre), is singleton. As was proved in [13] and [14], X is a topological IFS fractal iff X is homeomorphic to the attractor of some weak IFS (in particular, it is metrizable).…”

On Fryszkowski’s problem

“…For an application of Bessaga's converse see [20] and for some other converses of the contraction principle see [3], [7], [9], [12] and [17]. For more results along this line of research one can consult [1], [8], [13], [14], [15] and [23].…”