Emergent scalar symmetry in discrete dynamical systems

Abstract

Discrete quadratic dynamical systems are frequently used to define models of reality. To understand these recursive systems is to know the behavior of their complex orbits. The Mandelbrot Set plays an important role in this: the behavior of the orbits of its points can be extrapolated to all quadratic polynomials. In calculating the map of periods for the orbits of the points belonging to the Mandelbrot Set by means of discrete (and finite) encoding, some singular points called symmetrical points appear on the map. Such points configure, in a surprisingly harmonic and regular manner, the values of the rest of points in the map of periods. Thus, knowing these points and their properties is extremely helpful to better understand the behavior of quadratic and discrete dynamical systems. Important properties of such points are described in this article. We emphasize the emergent scalar symmetry property, which appears as a consequence of the finiteness of discrete values with which we are inevitably limited to represent the continuum. Thanks to this property, the images created around these points are not altered when removing or selecting one of their several pixel rows and columns. The image can be sampled in its vicinity, at any scale, without losing information regarding the vicinity. We propose a justification on why the map of periods is highly hypersensitive to small changes in the parameters defining the calculation mode of the map.