Graphical exploration of the connectivity sets of alternated Julia sets; M, the set of disconnected alternated Julia sets

TL;DR

This paper uses computer graphics to visualize the connectivity domains of alternated Julia sets, revealing their complex four-dimensional fractal structures and the relationship with the Mandelbrot set.

Contribution

It extends analytical results by visualizing the connectivity properties of alternated Julia sets, highlighting their four-dimensional fractal nature and the connection to the Mandelbrot set.

Findings

Connectivity domains are four-dimensional fractals.

Mandelbrot set includes parameters for both connected and disconnected alternated Julia sets.

Visualizations support theoretical properties of alternated Julia sets.

Abstract

Using computer graphics and visualization algorithms, we extend in this work the results obtained analytically in [1], on the connectivity domains of alternated Julia sets, defined by switching the dynamics of two quadratic Julia sets. As proved in [1], the alternated Julia sets exhibit, as for polynomials of degree greater than two, the disconnectivity property in addition to the known dichotomy property (connectedness and totally disconnectedness) which characterizes the standard Julia sets. Via computer graphics, we unveil these connectivity domains which are four-dimensional fractals. The computer graphics results show here, without substituting the proof but serving as a research guide, that for the alternated Julia sets, the Mandelbrot set consists of the set of all parameter values, for which each alternated Julia set is not only connected, but also disconnected.