About Structure of a connected Quaternion-JUllA-Set and Symmetries of u related JULIA-Network

Abstract

Udo E. Steinemann

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.

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