Micro and Macro Fractals generated by multi-valued dynamical systems

Abstract

Given a multi-valued function $Φ$ on a topological space $X$ we study the properties of its fixed fractal, which is defined as the closure of the orbit $Φ^ω(Fix(Φ))=\bigcup_{n\inω}Φ^n(Fix(Φ))$ of the set $Fix(Φ)=\{x\in X:x\inΦ(x)\}$ of fixed points of $Φ$. A special attention is paid to the duality between micro-fractals and macro-fractals, which are fixed fractals for a contracting compact-valued function $Φ$ on a complete metric space $X$ and its inverse multi-function $Φ^{-1}$. 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 Sierpinski triangle, Sierpinski 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.