arXiv subjects
Publications and source records attributed to Natalia Novosad.
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.
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.