Quadratic dynamics in binary number systems

Abstract

We describe the quadratic dynamics in certain two-component number systems, which like the complex numbers, can be expressed as rings of two by two real matrices. This description is accomplished using the properties of the real quadratic family and its derivative. We also demonstrate that the Mandelbrot set for any of these systems may be defined in two equivalent ways that are analogous to the two characterizations of the usual complex Mandelbrot set.

Quadratic Dynamics Over Hyperbolic Numbers

“…For hyperbolic parameters outside the hyperbolic Mandelbrot set, the filled Julia set has 3 possible topological descriptions, if it is not empty, in contrast to the complex case where it is always a non-empty totally disconnected set. These results were proved in [1,2,4,5,6,7] and are reviewed here.…”

“…Then there are only three ways of combining them if neither is empty: both connected, both totally disconnected or one connected and the other totally disconnected. The following topological description of the hyperbolic filled Julia set was proved in [2,4,7]. Computer generated images are given in [2].…”

Iteration of Quadratic Maps on Coquaternions

“…From the study conducted by Fishback [11] we can conclude that the behavior, in terms of stability, of the above points accompanies the behavior of the corresponding real parts: the fixed point 11…”

“…More recently, the study of the dynamics of this map restricted to the other two cycle planes has also drawn the attention of several authors; see, e.g. [2,10,11,17]; a discussion based on the use of 2 × 2 matrices can also be seen in [18]. In the rest of this section we present a very brief discussion of the results on the dynamics of f c in each of the cycle planes which are relevant to our future work.…”

Basins of attraction for a quadratic coquaternionic map

“…Since the parameter c is a regular parameter 5 of the real map f , then, for any x 0 in the real basin of attraction of this cycle, we have lim n→∞ (f n ) (x 0 ) = 0; see e.g. [12]. This means that a coquaternion of the form q = x 0 + αi + αj, with x 0 ∈ B R (C ) and α an arbitrarily chosen real number, belongs to B(C ), showing that this basin is not bounded.…”