Finding Dynamics for Fractals

TL;DR

This paper explores the idea of considering fractals, like Julia and Mandelbrot sets, as states and trajectories in dynamical systems to deepen understanding of chaos and the universe's structure.

Contribution

It introduces a novel perspective of treating fractals as points and trajectories within dynamical systems, aiming to integrate fractal structures into the study of dynamics.

Findings

Fractals can be mapped as states in dynamical systems.

Julia and Mandelbrot sets are used as initial points for trajectories.

The approach suggests fractals are dense in the universe.

Abstract

The famous Laplace's Demon is not only of strict physical determinism, but also related to the power of differential equations. When deterministically extended structures are taken into consideration, it is admissible that fractals are dense both in the nature and in the dynamics. In particular, this is true because fractal structures are closely related to chaos. This implies that dynamics have to be an instrument of the extension. Oppositely, one can animate the arguments for the Demon if dynamics will be investigated with fractals. To make advances in the direction, first of all, one should consider fractals as states of dynamics. In other words, instead of single points and finite/infinite dimensional vectors, fractals should be points of trajectories as well as trajectories themselves. If one realizes this approach, fractals will be proved to be dense in the universe, since modeling the real world is based on differential equations and their developments. Our main goal is to initiate the involvement of fractals as states of dynamical systems, and in the first step we answer the simple question "How can fractals be mapped?". In the present study Julia and Mandelbrot sets are considered as initial points for the trajectories of the dynamics.