Micro and Macro Fractals generated by multi-valued dynamical systems

TL;DR

This paper explores the properties of fixed fractals generated by multi-valued dynamical systems, focusing on the duality between micro- and macro-fractals, and provides algorithms to visualize macro-fractals related to well-known micro-fractals.

Contribution

It introduces the concept of fixed fractals for multi-valued functions, analyzes the duality between micro- and macro-fractals, and develops algorithms for visualizing macro-fractals.

Findings

Macro-fractals exhibit large-scale fractal structures.

Duality between micro- and macro-fractals is established.

Algorithms enable visualization of macro-fractals related to classical fractals.

Abstract

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