1. MICRO AND MACRO FRACTALS GENERATED BY MULTI-VALUED DYNAMICAL SYSTEMS.
- Author
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BANAKH, T. and NOVOSAD, N.
- Subjects
FRACTALS ,TOPOLOGICAL spaces ,DUALITY theory (Mathematics) ,METRIC spaces ,SNOWFLAKES ,DYNAMICAL systems - Abstract
Given a multi-valued function Φ : X ⊸ X on a topological space X we study the properties of its fixed fractal, which is defined as the closure of the orbit Φ
ω (*Φ ) = ⋃n∈ω Φn (*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 and 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. [ABSTRACT FROM AUTHOR]- Published
- 2014
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