Mandelbrot fixed systems… Some questions.

Within the set functions of the form Z=Z^2+C decay to a limited number of repeating values. The starting value for these interations was Z0 = 0+0i and the system was run for 1E5 iterations before testing for repetition.

Here Fixed-Points are white (a single value). Binary systems with each value mapped to the other value by the function in Red. Triple systems in Blue and Quaternary systems in Green.

Are there any values of C forming systems that never repeat?

Looking at the arrangement of “dusts” there is a definite pattern. It appears to involve some level of reflection and rotation. The sense I formed is like walking into a hallof mirrors with the object image being broken up and distorted by the function. Is this pattern fractal and exist at lower scales?

Regarding fixed points. So the repetative use of the function gradually removes information from the system until the path collapses into a repeating system. Reversing the function now cannot retrace the path. The information is lost.

Given a fixed point how many points in the set collapse to this point? Is there a method to reverse it?

The existence of dynamic systems with more than 1 state suggests that the SRH should not be concerned with the point itself, but rather the system of values/terms which provide something like a “hermaneutic circle”. Given 1 point we can travel to all the others, but we cannot leave the system. In Russel’s Principia rather than just a single statement which maps to itself via Godel Numbering we can have any number of closed loops. A contradiction can be sort in any of these loops by creating a mirror image of the loop using negation. Because the system is defined by the logic of the loop, the negative mirror image can be easily constructed to contradict the logic. I am provable/I am not provable.

 

Julia Sets

For the function Z = Z^2+(0.4 + 0.6i) the fixed point Zf occurs where Zn+1 == Zn. There are always 2 solutions for every function.

Zf1 = 0.983976  - 0.619865 i

Zf2 = 0.0160238 +0.619865 i

Plotted on the Julia set:

Interestingly the symetrical points in the other 2 quadrants iterate immediately to these 2 fixed points.

Clearly fixed points are integral to the set, marking here the centre of the cross and the end of the long branch. However unclear what special position they hold in the structure.

The original guess was that fixed points are strange attractors. They determine the anchors for the structures. Perhaps this is still partly true.

Logistic Equation

This is much simpler to explore. The question is why does repeated application of a function decay it. The suggestion is decrease of entropy and loss of information when we apply a function. To process of mapping increases the order but uses energy. And under what conditions does it decay to a fixed, binary etc system. Is a chaotic system “really” chaotic does it not increase order at all, or is it just very slow.

Remember the system bouncing between the Quadratic and the y=x line as a way to visualise the process. Why do only certain curves create bifurcation.

SRH revisited (Dynamic Systems, Affine transformation & Mandelbrot)

 

All affine transformations represented by A(r) = p+\[Sigma]Mr. We can use this to explore 2D fixed points of repeating systems. p provides translation, \[Sigma] provides scaling, M provides rotation.

The generals SRH question is: what conditions lead to SRH? i.e. When can you not draw conclusions about a scope. e.g. All statements are false. The scope includes this statement -> repeating system -> SRH

Repeating systems cause Fixed points. But fixed points -> contradictions. Godels Theorem based on a Fixed Point (this statement is provable) which easily turned into a contradiction. Because it is fixed, then it is unconditional, and then a mirror image of this Fixed Point is also unconditional and fixed, but also contradictory. Even Black Holes are fixed points arriving from repeated application of gravitational laws within closed systems.

The SRH states that there can be no unconditional truth.

In this example we see that the translation vector p provides the conditional "data" which is transformed to the Fixed Point F in 1 step  by subtracting the negative rotation from itself. While the forward affine transformation f is an infinite regress (when s < 1) and infinite digress when s > 1 and a loop when s = 1. Thus the fixed point is determined by the transformation conditions {p, \[Sigma], \[Theta]} while the input starting point in the plane is irrelevant. The transformation itself determined the result in the repeated system. i.e. repeated system lose information. This must mean their entropy increases. And the only thing left is the transformation itself which is refreshed in each sycle (by definition of apply a fixed transformation). The "Fixed" here applies to the fixed nature of the transformation.

Check this... perhaps \[Sigma] > 1 and p < 1 can regress also...

Note. in the Mandelbrot a plot of the paths for each starting C produces the Buddhabrot density map. So the map of information being lost is interesting. Buddha & entropy & impermanence. Boltzman must be wrong but subtly.

On the real line the constant Mandelbrot system (z = z^2 + c) loses information to the fixed points {0,1,\[Infinity]}. So cx loses almost all its info.

The famous set is the map of rate the system moves to fixed point \[Infinity] with c. It ignores 0.

But fixed points are just 1 type of fixed systems. Not all points in the map c share common points. Many share common cycles. If a system decaying arrives at a value in one of these cycles it is trapped in the cycle. The Buddhabrot plots the density of points in the these systems i.e. the points in c that are most visited during decay.

As the fixed in fixed point arrives from the fixed nature of the repeating transformation (not the input data) so the fixed loops reflect the fixed nature of the transformation. So we should refer not just to fixed "points" but to the broader class of "fixed system".

2DO
In the Mandelbrot set how many fixed systems are there? As the points decay when they arrive back in their own decay path we form a loop. That loop is recorded and coloured and the starting point gets this colour. The final map will show which loops each point arrive in, and also the diversity of loop systems.

2DO
I believe there are some extraordinary values of p under certain transformations of {\[Theta]} which diverged with \[Sigma] < 1.

If we take the unit area (map) i.e. {x,y} | 0<x<1 and 0<y<1 then SRH is particularly interested in those M'=F(M) which fall within M i.e. where M' \[And] M != {}. Altho M' can occupy the whole real plane without issue. Is the unit area of no real interest? Does SRH showing interest in "self" reference actually missing the point. The point (fixed point ;-) .. is SRH seeking a fixed point of Truth -> contradiction!!! i.e. if SRH is true than it provides a universal fixed point -> can construct a contradiction by symettry + negation = reflection) .. the point is that the system takes its own output... that is the self rather than the constraints on the input?  

2DO
1) Also consider chaos. Why is there no chaos in affine real plane? Is this true.
2) How does Mandelbrot transformation differ from affine transformation. z^2 is rotation and scaling and c is translation. Can we recreate Mandelbrot as affine transformation -> thus find chaos.
3) Why do repeating/dynamic system provide so little stability. ONly with \[Sigma] = 1 can we find stability. Is this true?