Advances in Machine Learning & Artificial Intelligence(AMLAI)
ISSN: 2769-545X | DOI: 10.33140/AMLAI
Research Article - (2024) Volume 5, Issue 2
About Structure of a connected Quaternion-JUllA-Set and Symmetries of u related JULIA-Network
Received Date: Mar 12, 2024 / Accepted Date: Apr 23, 2024 / Published Date: Apr 29, 2024
Abstract
If a variable is replace by its square and subsequently enlarged by a constant during a number of iteration-steps in quaternion-space, a network of (3) sets will be built gradually. As long as for the iteration-constant certain conditions are fulfilled, the network will consist of: an rrnbounded set (escape-set) with trajectories escaping to infinity during course of the iteration, a bounded set (prisoner-set) with trajectories tending to a sink-point and a further bounded one (JULIA-set) with a fixed-point as repeller having a repulsive effect on all points of both the other sets. The iteration will continue until the attracting sink-point of prisoner-set and the repeliing fixedpoint on JULIA-set have been found. This situation is reached if predecessor- and successor-state of the iteration became equal. The fixed-point-condition provisionally formulated in general terms of quaternions, can be separated into (3) sub-conditions. When heeding the HAMILTONian-rules for interactions of the imaginary sub-spaces of the quaternion-space, each sub-condition will be appropriate for one imaginary subspaces and independently debatable. Knowledge of fixed-points from this fundamental network will one enable to study the structure of a connected JULIA-set.
The Iteration will start from (1) on real-axis, this is not a restriction on generaiity because an appropriate scaling on real- axis can always be archived this way. It will become obvious, that the fixed-points in prisonerand JIILIA-set will depend on the iteration-constant only. Thus (16) different constants chosen appropriately will enable to arrange (16) fixed-points of JLTLIA-sets in the square-points of a hyper-cube and thereby together with the JULIA-sets to built a related JULIA-network. The symmetry-properties of this related .IULIA-network can be studied on base of a hyper-cube's symmetry-group extended by some additional considerations.
Introduction
In the following attention is appiied to the results of an iteration, which takes place in quaternion-space (a space of hyper-cubes with its space-elements) a layout of this is given next:
Each hyper-cube: • Is surrounded by (8) cubes each one with (6) surfaces. Thus all together, cubes will have (48) surfaces. • Because the cubes wiil slmre surfaces, onlv (24) surfaces will have to be counted effectively. The quaternion-space is spanned by a real unit-vector (e) vertical to a tripod of imaginary unit-vectors {i ^j ^d,}. Among these referencevectors t}re HAMILTONian rules must hold:
References
1. Peitgen, H. O., Jurgens, H., & Saupe, D. (1992). Chaos and Fracal: New Frontiers of Science Springer. New York, p. 9e84.
2. Douady, A., Hubbard, J. H., & Lavaurs, P. (1984). Etude dynamique des polynômes complexes.
3. Mandelbrot, B. B. (1980). Fractal aspects of the iteration of z * Xz(1-z) for complex \ and z. Annals of the New York Academy of Sciences, 357.
Copyright: © 2025 This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.