Micro and Macro Fractals generated by multi-valued dynamical systems

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