Micro and Macro Fractals Generated by Multi-Valued Dynamical Systems

Abstract

Abstract. Given a multi-valued function Φ : X X on a topological space X we study the properties of its fixed fractal Φ , which is defined as the closure of the orbit Φ ω ( * Φ ) = n∈ω Φ n ( * Φ ) of the set * Φ = {x ∈ X : x ∈ Φ(x)} of fixed points of Φ. A special attention is paid to the duality between micro-fractals and macro-fractals, which are fixed fractals Φ and Φ −1 for a contracting compact-valued function Φ : X X on a complete metric space X. With help of algorithms (described in this paper) we generate various images of macro-fractals which are dual to some well-known micro-fractals like the fractal cross, the Sierpiński triangle, Sierpiński carpet, the Koch curve, or the fractal snowflakes. The obtained images show that macro-fractals have a large-scale fractal structure, which becomes clearly visible after a suitable zooming.

Central Open Sets Tilings

“…The fast basin is the set of points such that some finite orbit meets A. Fast basins are related to but distinct from the fractal blow-ups introduced by Strichartz [24] and the macro-fractals introduced by Banakh and Novosad [2]. We will use B\A in calculations in Section 4.…”

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

“…h C where ðI n Þ n2N (respectively, ðJ n Þ n2N ) are the steps in the construction of Cantor set C. Barnsley [6] calls the space ðP k ðXÞ; hÞ, the life space of fractal. Recently, Banakh and Novosad [5] proposed a fractal approach using Vietoris topology (in a more general setting than the one used for the Hausdorff topology).…”

“…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]). …”