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?